BURGERS TURBULENCE AND RANDOM LAGRANGIAN SYSTEMS

RENATO ITURRIAGA, KONSTANTIN KHANIN

Abstract. We consider spatially periodic inviscid random forced Burgers equation in arbitrary dimension and the random time-dependent Lagrangian system related to it. We construct a unique stationary distribution for “viscosity” solutions of the Burgers equation. We also show that with probability 1 there exists a unique minimizing trajectory for the random Lagrangian system. These minimizing trajectories generate a unique minimizing measure for the non-random skew-product extension of the Lagrangian system.

1. Introduction

In this paper we discuss two closely related problems: stationary solutions for ddimensional spatially periodic inviscid random forced Burgers equation and properties of minimizing orbits and measures for random time-dependent Lagrangian systems. The random forced Burgers equation was a subject of intensive study in the physical literature in the last 5 years (see [9, 40, 7, 25, 6, 24, 3, 5, 23]). The methods used vary from numerical simulations to quantum field theoretic closure techniques. Although Burgers equation arises naturally in many different physical problems, recent interest was mostly motivated by the hydrodynamics applications and the theory of turbulence. The mathematical theory in the one-dimensional case was developed in [14, 15]. It was proven that there exists a unique stationary distribution for the solutions of the random inviscid Burgers equation, and typical solutions are piecewise smooth with finite number of jump discontinuities corresponding to shocks. The analysis in [14, 15] was based on the study of geometric and dynamical properties of minimizing orbits. It was shown that there exists a unique global minimizer for the corresponding Lagrangian system. Moreover, the global minimizer is a hyperbolic orbit of the Lagrangian flow, and its unstable manifold is closely connected with the solutions of inviscid Burgers equation. One can say that all shocks, except a special one which is called the main shock, are connected with local double-folds of the unstable manifold. On the contrary, the main shock has a topological nature. Its existence follows from spatial periodicity of the equation. It is important to mention that the geometrical picture proved in [14] allows not only analyse the structure of singularities for typical stationary solutions, but also leads to quantitative predictions for universal scaling exponents related to the pdf (probability distribution function) for the velocity gradients ([15]).

One of the main aims of the present paper is to study what happens in the ddimensional case. The methods which were used in [14, 15] were purely one-dimensional and in order to achieve our goals we use in much more systematic way the Lagrangian formalism and the Hamilton-Jacobi equation. This brings us close to a connected line of research: study of minimizing orbits for Lagrangian systems. The theory of minimizing orbits for Lagrangian systems with autonomous or time-periodic potentials has a long history. It goes back to the works of M. Morse and G. Hedlund (see [38, 26])). In the last 15 years this subject became a very active research area in connection with Aubry-Mather theory (see [2, 36, 37, 34, 35, 20, 12]). R. Mañé has conjectured that for a generic time-periodic potential there exists a unique global minimizer which is a hyperbolic periodic trajectory of the Lagrangian system. This conjecture remains open. However, a related result on the uniqueness of a minimizing measure has been proved in [34]. It was also shown in [11] that the invariant measure is generically hyperbolic provided it is supported by a single periodic orbit.

The work presented here can be considered as a theory of minimizing orbits for random Lagrangian systems. This problem is quite interesting on its own even without reference to the hydrodynamics and the Burgers equation. It is worthwhile to emphasise a connection between the random case and Mañé’s setting. Indeed, random Lagrangian systems are generic with probability 1 in the measure-theoretic sense, while Mañé’s conjecture is related to topologically generic Lagrangian systems. Not surprisingly, as we shall see there is also certain similarity in the results in both cases.

We start with the d-dimensional Burgers equation:

∂_{t}u + (u · ∇)u = νΔu + f(y, t)  (1)

where u(y, t) = (u_{i}(y, t), 1 ≤ i ≤ d) is a velocity field, y = (y_{i}, 1 ≤ i ≤ d) ∈ ℝ^{d}, and f(y, t) = (f_{i}(t, y), 1 ≤ i ≤ d) is an external force. As we have already mentioned above Burgers equation is a prototype equation which naturally appears in very different physical problems. It can be obtained from the Navier-Stokes equations

∂_{t}u + (u · ∇)u = νΔu − ∇p + f(y, t), div u = 0.  (2)

by dropping the pressure term −∇p and incompressibility condition div u = 0. Sometimes it is said that Burgers equation describes pressure-less turbulence. Correspondingly it is connected with the dynamics of low density gases. Since we are mostly interested in hydrodynamics motivation and a turbulent type behaviour, we shall assume that the equation is driven by a potential random force f(y, t) = −∇F (y, t), which is smooth in the space variable y and quite irregular in time. This basically means that the forcing term does not excite high harmonics. We shall also assume that the random potentials F (y, t) are spatially-periodic. The irregular behaviour in time will be captured by two main models. In the first model, which we call “white force”, the forcing term is similar to white noise in time. The second model corresponds to a “kicking force” for which the velocity field u changes discontinuously after kicks. We shall assume that the kicks occur at discrete moments of time. The forcing term is responsible for the constant pumping of energy into the system. The energy also dissipates through different dissipation mechanisms. The balance between pumping and dissipation should in principle lead to the establishment of a stationary probability distribution for solutions. This is a general philosophy and the first task of a mathematical theory of turbulence consists in establishing the existence and, preferably, uniqueness of this stationary distribution, or, in dynamical systems terminology, of an invariant measure. The difficulty of the problem depends on the mechanism of dissipation involved. This mechanism obviously depends on the dimension d. The difficulty of the problem increases when one passes from from viscous case ν > 0 to inviscid limit ν → 0, and from dimension d = 1 to higher dimensions.

It is important to mention, that the construction of the stationary distribution or the invariant measure corresponds to the limit t → ∞. At the same time, the inviscid case, which is mostly interesting for turbulence theory, requires another limiting process ν → 0. It is tempting, first, to construct an invariant measure μ_{ν} for ν > 0, that is first to take limit as t → ∞ and then study limit of μ_{ν} as ν → 0. However, it is extremely difficult to control the last limiting process. Experience of the random Burgers equation, for which invariant measure can be constructed for both viscous (see [42]) and inviscid cases, suggests that it is more productive to consider from the very beginning ν = 0, construct unique invariant measure μ, and only later prove that μ_{ν} → μ as ν → 0.

As we have mentioned above, a mathematical theory starts with establishing the existence and uniqueness of an invariant measure. However, from the physical point of view the most interesting problem is the analysis of statistical properties of stationary solutions. In other words, how a typical solution looks like? What are the leading singularities? How the power spectrum decays? What are the asymptotic properties of pdf’s and the structure functions? Those questions are much harder to answer. They form the core of the problem of turbulence. It is probably fair to say, that the problem of turbulence splits into “soft part” of existence-uniqueness statements and a “hard part” of analysing properties of typical solutions. In this paper we show that for the d-dimensional spatially-periodic inviscid Burgers equation one can complete the “soft part” in a very general situation, and also answer some of the “hard part” questions. Notice that the difficulties of the multi-dimensional case are connected with much more complicated structure of singularities for typical solutions. We shall discuss it in more detail in the last section of the paper.

Coming back to the problem of Lagrangian minimizers we can roughly formulate our main results in the following way: with probability 1 there exists a unique global minimizer for random Lagrangian system. This result holds for arbitrary compact connected Riemannian manifold. However, in this paper we consider for simplicity only the case of d-dimensional torus 𝕋^{d}, which corresponds to the case of spatiallyperiodic potentials. As we shall see, the case of “kicking” force corresponds to a random generalisation of Aubry - Mather theory (see [2, 36]). The Lagrangian in this case can be considered as a random Frenkel-Kontorova type model, which describes an infinite chain of particles placed in periodic potentials and connected by elastic springs. In the random situation the potentials are chosen independently for each particle, contrary to the classical case when one periodic potential serves all of them. In the non-random case the structure of global minimizers is very complicated. It crucially depends on the arithmetic properties of the rotation vector. In the case of fixed Diophantine rotation number (d = 1) it is believed that one has transition from elliptic minimizers for small potentials to hyperbolic global minimizers when potentials are large enough. This transition is related to destruction of invariant KAM curves which correspond to elliptic minimizers. All this complicated picture should be compared with the stated above uniqueness of the global minimizer in the random situation. Quite naturally, the number-theoretical properties of the rotation vector play no role in the random case.

The paper has the following structure. In Section 2 we define the random force and discuss the notion of invariant measure and stationary distribution for the random Burgers equation. In Section 3 we introduce the random Hamilton-Jacobi equation and discuss some properties of the minimizers. The main results are formulated in Section 4. In Section 5 we prove uniqueness statements for solutions of random Hamilton-Jacobi equation in infinite domains. Section 6 is devoted to the stationary distribution for the random Burgers equation. We first construct the stationary distribution and then prove its uniqueness. In Section 7 we show that with probability 1 the global minimizer is unique. We finish with concluding remarks in Section 8. In the Appendix we prove technical lemmas on the stochastic Lagrangian flow. We have deliberately decided to state the results only after all the setting and preliminaries are completed.

However, the experts who are familiar enough with the background will be able to understand the results in Section 4 without introductory Sections 2 and 3.

2. Random setting

To make the general discussion more precise we have to make concrete assumptions on the random force f(y, t) = −∇F (y, t). As we have already mentioned above, all the results of this paper can be extended to Burgers equation and Lagrangian systems on a compact connected Riemannian manifold. However, for simplicity we have chosen to deal only with the case of the torus 𝕋^{d} = ℝ^{d}/ℤ^{d} which corresponds to periodic potentials F (y, t) = F (y + m, t), m ∈ ℤ. We shall denote points of 𝕋^{d} by x and points of the universal cover ℝ^{d} by y.

We start with the “white force” case:

f(y, t) = −∇F (y, t), F (y, t) = ∑_{k=1}^{N} F^{k}(y)B_{k}(t).  (3)

Here F^{k}(y) are smooth periodic potentials, F^{k}(y) = F^{k}(y + m), m ∈ ℤ and B_{k}(t) = Ẇ_{k}(t) are independent white noises, corresponding to independent Wiener processes W_{k}(t). Summation in (3) can be finite or infinite. In the latter case one has to make sure that potentials F^{k}(y) decay fast enough so that the series converges and defines a smooth forcing term f(y, t). For simplicity everywhere below we assume that N is finite. We also assume that all potentials F^{k}(y) are C^{∞}−smooth although all the results below hold true in the case when F^{k} belong to C^{5}. Burgers equation (1) with the random forcing (3) becomes stochastic PDE. However, one can essentially eliminate stochastic part by considering “kicking force” model. Let

f(y, t) = −∇F (y, t), F (y, t) = ∑_{j∈ℤ}F_{j}(y)δ(t − t_{j}).  (4)

The force in (4) corresponds to the situation when the system evolves without any force between moments of kicks t_{j−1} and t_{j}, so that u(y, t_{j}−) is obtained from u(y, t_{j−1}+) as a result of free evolution of a system. Then, suddenly, we apply a kicking force which change solution discontinuously, i.e.

u(y, t_{j}+) = u(y, t_{j}−) − ∇F_{j}(y).  (5)

Smooth periodic potentials {F_{j}(y), j ∈ ℤ} form a realization of some stationary potential-valued random process. The simplest situation corresponds to Bernoulli process, when different F_{j} are chosen independently according to some probability distribution χ ∈ P(C^{1}(𝕋^{d})) in a space of smooth potentials. Here and below we denote by P(M) the set of all probability distributions on the measure space M. One can also make different assumptions on kicking times t_{j}. We shall just indicate two natural cases. The first one corresponds to periodic sequence t_{j} = hj, j ∈ ℤ for some positive h in ℝ. In the second situation kicking times are determined by the realization of a stationary random process which is independent from the random process F_{j}(y). For example, {t_{j}, j ∈ ℤ} can be a realization of Poisson process. In this paper we consider only the simplest case when kicking potentials form Bernoulli sequence, and kicking times are periodic with h = 1.

In both “white” and “kicked” settings the force is given by a stationary random process. Denote by (Ω, F, P) the probability space corresponding to this random process. We can assume without loss of generality that (Ω, F, P) is a complete measure space. Sometimes we shall use the notation f^{ω}(y, t), F^{ω}(y, t), F_{j}^{ω}, ω ∈ Ω, for the forcing term and the potentials, indicating their randomness. The time shift t → t + τ corresponds to the flow of automorphisms of the probability space Ω :

θ^{τ} : Ω → Ω, θ^{τ}P = P, F^{ω}(y, t + τ) = F^{θ^{τ} ω}(y, t), τ ∈ ℝ,  (6)

Equality θ^{τ}P = P in (6) expresses the stationarity of the random force. Obviously, in the case of periodic kicking times one should consider only the values τ = j, j ∈ ℤ in (6). We can define now a standard skew-product structure which leads to the definition of invariant measure. Consider the Cauchy problem for the random Burgers equation (1). Denote by U some natural functional space, which is invariant under the evolution given by (1), or an invariant set in such space. This means that u(y, t) ∈ U for all t > 0, provided u(y, 0) ∈ U. Denote by B^{ω}(τ) a nonlinear random transformation on U corresponding to the solution of the Cauchy problem on the time interval [0, τ] with the random force f^{ω}(y, t). In other words, u(y, τ) = B^{ω}(τ)u(y, 0) in the “white force” case and u(y, τ+) = B^{ω}(τ)u(y, 0+), τ ∈ ℤ, in the “kicked” case. Clearly, solution of the equation for longer times implies iteration of B^{ω}(τ) for shifted ω. It is easy to see that this process corresponds to non-random transformation B̂ acting on the product space U × Ω:

B̂(τ)(u, ω) = (B^{ω}(τ)u, θ^{τ}ω).  (7)

Definition 1. A probability measure ν ∈ P(U × Ω) is called an invariant measure for the random equation if

1. For any τ ∈ ℝ (τ ∈ ℤ in the “kicked” case), B̂(τ) preserves ν : B̂_{∗}(τ)ν = ν.

2. Marginal distribution of ν on Ω is given by P, i.e.

ν(du, dω) = ν^{ω}(du)P (dω),

where ν^{ω}(du) are the conditional distributions on U under the condition of fixed ω.

3. The conditional distributions ν^{ω} are measurable with respect to the σ - algebra F_{−∞}^{0} generated by the random process on the time interval (−∞, 0].

The last condition means that ν^{ω} does not depend on the future behaviour of a random force f^{ω}(y, t), t > 0. It may look a bit artificial but one may say that all physically relevant invariant measures always satisfy this condition.

We shall describe now a slightly different point of view which also leads to the notion of a stationary distribution for solutions of the random equation. It is easy to see that B^{ω}(τ) defines a stationary Markov process on U with transition probabilities given by:

p_{τ}(A | u_{0}) = P (ω : u(y, τ) = B^{ω}(τ)u_{0} ∈ A) .  (8)

Definition 2. A probability measure μ ∈ P(U) is called a stationary distribution for the random equation if it is a stationary distribution for the Markov process (8), i.e. for all τ > 0 (τ ∈ ℕ in the kicked case):

μ(A) = ∫_{U} p_{τ}(A | u_{o})μ(du_{0}) .

The notions of invariant measure and stationary distribution for a random equation are closely connected. Namely, if ν is an invariant measure then its marginal distribution μ(du) = ∫_{Ω} ν^{ω}(du)P (dω) is a stationary distribution for the corresponding Markov process. Notice that this property holds only if an invariant measure ν satisfies condition 3 in Definition 1. The opposite statement is also correct. It follows from [1, 13] that any stationary distribution μ(du) can be obtained as the projection of a unique invariant measure ν(du, dω) = ν^{ω}(du)P (dω) satisfying condition 3.

3. Random Hamilton-Jacobi equation and minimizers

It is well known that due to the creation of shocks the inviscid Burgers equation has no strong, or, in other words, smooth solutions. However, there exists a unique “physical” weak solution for the Cauchy problem, which is called the viscosity solution. Given initial data u(x, 0) = u_{0}(x) one can construct the viscosity solutions by considering the viscous Burgers equation and taking the limit as ν → 0. We shall always assume that u_{0} is a gradient-like function, i.e. u_{0} = ∇ϕ_{0}, where ϕ_{0} is a Lipschitz function. It is easy to see that the Burgers equation with a potential force is an evolution equation in a space of gradient-like functions. This means that the set of gradient-like functions is an invariant set. Taking u(x, t) = ∇ϕ(x, t) we obtain the Hamilton-Jacobi equation for ϕ:

∂_{t}ϕ(y, t) + \frac{1}{2}(∇ϕ(y, t))^{2} + F^{ω}(y, t) = 0.  (9)

As in the case of the Burgers equation there are many weak solutions to the Hamilton-Jacobi equation. However, there exists a unique viscosity solution corresponding to the Burgers viscosity solution. The exact expression for the viscosity solution is given by the Hopf-Lax-Oleinik variational principle. The variational principle was first proved in the one-dimensional case ([27, 32, 39]), and then extended to the multi-dimensional Hamilton-Jacobi equation by P. Lions ([33], see also A. Fathi [20] for the case of general Riemannian manifolds). It says that the viscosity solution of the Cauchy problem for (9) in ℝ^{d} × [0, T] with the initial condition ϕ(·, 0) = ϕ_{0} is given by:

ϕ(y, t) = inf [ϕ_{0}(γ(0)) + ∫_{0}^{t} (\frac{1}{2}γ̇^{2} − F^{ω}(γ(τ), τ)) dτ] , t ∈ [0, T],  (10)

where the infimum is taken over all absolutely continuous curves γ : [0, t] → ℝ^{d} such that γ(t) = y.

If the potentials F^{ω} are ℤ^{d}-periodic functions, and the initial value ϕ_{0}(y) can be written in the form: ϕ_{0}(y) = b · y + ψ_{0}(y), where b ∈ ℝ^{d} and ψ_{0}(y) is ℤ^{d}-periodic, then the solution (10) preserves the same form. Namely,

ϕ(y, t) = b · y + ψ(y, t), t > 0,  (11)

where ψ(y, t) is a ℤ^{d}-periodic function of y. The linear form b·y can be considered as a first integral of the Hamilton-Jacobi equation. For the Burgers equation b corresponds to the average velocity

b = ∫_{𝕋^{d}} u(y, t)dx ,

which is well known to be a first integral. From now one we shall assume that the value of the first integral b is fixed.

Using formula (10) we get the variational principle for ψ(x, t), which we consider now as a function on the torus 𝕋^{d}. For fixed b denote the action of a curve γ : [s, t] → 𝕋^{d} by

A_{s,t}^{ω,b}(γ) = ∫_{s}^{t} (\frac{1}{2}(γ̇(τ) − b)^{2} − F^{ω}(γ(τ), τ) − \frac{b^{2}}{2} ) dτ.  (12)

Then,

ψ(x, t) = inf_{γ∈AC(0;x,t)} (ψ_{0}(γ(0)) + A_{0,t}^{ω,b}(γ)) ,  (13)

where AC(s; x, t) is the set of all absolutely continuous curves γ : [s, t] → 𝕋^{d} on the torus 𝕋^{d} such that γ(t) = x. We shall also use the following notations. Denote AC(x_{1}, s; x_{2}, t) the set of absolutely continuous curves γ : [s, t] → 𝕋^{d} such that γ(s) = x_{1} and γ(t) = x_{2}; AC(x, t) the set of absolutely continuous curves γ : (−∞, t] → 𝕋^{d} such that γ(t) = x; AC the set of all absolutely continuous curves γ : (−∞, +∞) → 𝕋^{d}. Finally, we define the Lax operator K_{s,t}^{ω,b} :

K_{s,t}^{ω,b}ψ(x) = inf_{γ∈AC(s;x,t)} (ψ(γ(s)) + A_{s,t}^{ω,b}(γ)) .  (14)

Obviously, a function ϕ(y, t), y ∈ 𝕋^{d}, t ∈ [t_{1}, t_{2}] is a “viscosity” solution of the Hamilton - Jacobi equation if ϕ(y, t) = b · y + ψ(y, t), where ψ is ℤ^{d}-periodic and for all t_{1} ≤ s < t ≤ t_{2} :

K_{s,t}^{ω,b}ψ(·, s) = ψ(·, t).  (15)

We next discuss an extension of the variational principle to the random case (3), (4) where dependence of the potentials on time is quite singular. In the case of “white force” the integral in (12) is in principle stochastic integral and it is not immediately clear why it is well defined for all absolutely continuous curves with probability 1. However, using integration by parts we can effectively eliminate the stochastic nature of the integral:

A_{s,t}^{ω,b}(γ) = ∫_{s}^{t} (\frac{1}{2}(γ̇(τ) − b)^{2} − ∑_{k=1}^{N} F^{k}(γ(τ))Ẇ_{k}(τ) − \frac{b^{2}}{2} ) dτ =

∫_{s}^{t} (\frac{1}{2}(γ̇(τ) − b)^{2} + ∑_{k=1}^{N} ∇F^{k}(γ(τ)) · γ̇(τ)(W_{k}(τ) − W_{k}(s)) − \frac{b^{2}}{2} ) dτ  (16)

− ∑_{k=1}^{N} F^{k}(γ(t))(W_{k}(t) − W_{k}(s)).

Everywhere below in the case of “white force” we shall use the action in the form (16). The proof of the fact that (11), (13), (16) indeed give the unique viscosity solution basically repeats the proof of the similar statement in [14], Appendix A.

In the “kicked” case the extension of the variational principle is straightforward. Let u(y, n±) = ∇ϕ(y, n±), ϕ(y, n±) = b · y + ψ(y, n±), ψ(y, n+) = ψ(y, n−) − F^{n,ω}(y). Denote by A_{m,n}^{ω,b}(X) the “kicked” action of the sequence X = {x_{i} ∈ 𝕋^{d}, m ≤ i ≤ n} :

A_{m,n}^{ω,b}(X) = ∑_{i=m}^{n−1} (\frac{1}{2}ρ_{b}^{2}(x_{i+1}, x_{i}) − F_{j}^{ω}(x_{i}) − \frac{b^{2}}{2} ) ,  (17)

where ρ_{b}(x, x′) = min_{k∈ℤ^{d}}(∥x−x′ −b−k∥). Notice that ρ(x, x′) = ρ_{0}(x, x′) is the metric on a flat torus 𝕋^{d}. Here and everywhere below we denote by ∥y∥ the Euclidean norm in ℝ^{d}. Then,

ψ(x, n−) = inf_{X∈S(0;x,n)} [ψ(x_{0}, 0−) + A_{m,n}^{ω,b}(X)] ,  (18)

where S(m; x, n) is the set of sequences X = {x_{i} ∈ 𝕋^{d}, m ≤ i ≤ n} such that x_{n} = x.

In analogy with the continuous case we shall also use the following notations: S(x′, m; x, n) is the set of all sequences {x_{i} ∈ 𝕋^{d}, m ≤ i ≤ n} such that x_{m} = x′, x_{n} = x; S(x, n) is the set of infinite sequences {x_{i} ∈ 𝕋^{d}, −∞ < i ≤ n} such that x_{n} = x; S is the set of all double-infinite sequences {x_{i} ∈ 𝕋^{d}, −∞ < i − ∞}. Denote by K_{m,n}^{ω,b} the Lax operator in the “kicked” case;

K_{m,n}^{ω,b}ψ(x) = inf_{X∈S(m;x,n)} (ψ(x_{m}) + A_{m,n}^{ω,b}(X)) .  (19)

Then, as in (15), for all m < n, m, n ∈ ℤ:

K_{m,n}^{ω,b}ψ(·, m−) = ψ(·, n−),  (20)

if ψ(·, n±) defines the viscosity weak solution to the “kicked” Hamilton-Jacobi equation. Notice, that the sequence ψ(·, n−) contains the whole information about a solution since ψ(x, n+) = ψ(x, n−) − F_{n}^{ω}(x).

Obviously, the Lax operators K_{s,t}^{ω,b}, K_{m,n}^{ω,b} commute with an operation of addition of a constant. This reflects the well known fact that potentials are defined up to an additive constant. By this reason it is convenient to define the action of Lax operators on the corresponding factor space. Denote by C(𝕋^{d})/ℝ the space of equivalence classes of continuous functions on 𝕋^{d} up to additional constant and by ψ̃ ∈ C(𝕋^{d})/ℝ the equivalence class of the potential ψ:

ψ̃ = {ψ_{1} ∈ C(𝕋^{d}) : ψ_{1} = ψ + const}.

We shall denote by K̃_{s,t}^{ω,b}, K̃_{m,n}^{ω,b} the actions of K_{s,t}^{ω,b}, K_{m,n}^{ω,b} on C(𝕋^{d})/ℝ. Let ψ̃(t), t ∈ I be a curve in C(𝕋^{d})/ℝ, where I is finite or infinite interval of time. We shall say that ψ̃(t), t ∈ I satisfies the factorized Hamilton-Jacobi equation for fixed b if for all s < t, s, t ∈ I:

K̃_{s,t}^{ω,b}ψ̃(s) = ψ̃(t).  (21)

As above, in the “kicked case” one has to replace s, t by m < n, m, n ∈ ℤ, K̃_{s,t}^{ω,b} by K̃_{m,n}^{ω,b} and ψ̃(s), ψ̃(t) by ψ̃(m−), ψ̃(n−).

We next define an important notion of a minimizer.

Definition 3. Consider the action A^{ω,b}.

1. A curve γ : [s, t] → 𝕋^{d} is called a minimizer if

A_{s,t}^{ω,b}(γ) = min_{σ∈AC(γ(s),s;γ(t).t)} A_{s,t}^{ω,b}(σ).

2. A curve γ : [s, t] → 𝕋^{d} is called a ψ−minimizer if

ψ(γ(s)) + A_{s,t}^{ω,b}(γ) = min_{σ∈AC(s;γ(t).t)} (ψ(σ(s)) + A_{s,t}^{ω,b}(σ)).

3. A curve γ_{x_{0},t_{0}} : (−∞, t_{0}] → 𝕋^{d} is called a one-sided minimizer if it is a minimizer for all (s, t), −∞ < s < t ≤ t_{0} and γ_{x_{0},t_{0}}(t_{0}) = x_{0}.

4. A curve γ : (−∞, ∞) → 𝕋^{d} is called a global minimizer if it is a minimizer for all (s, t), −∞ < s < t < ∞.

Similar definitions apply to the “kicked” case. One has just to replace curves γ by sequences X, the action A_{s,t}^{ω,b} by the “kicked” action A_{m,n}^{ω,b} and pairs (s, t) by the pairs of integer moments of time (m, n).

Clearly, minimizers are special trajectories of the Lagrangian flow L_{s}^{ω}, s ∈ ℝ which is generated by solutions of the Euler-Lagrange system of equations:

ẋ = v
v̇ = −∇F^{ω}(x, t).  (22)

Although formally equation (22) is a stochastic differential equation, as above, after integration by parts we can regard it as a non-stochastic integro-differential equation. Let x(t), t ∈ [t_{1}, t_{2}], be a C^{1} curve on 𝕋^{d}. We shall say that x(t) is a solution of (22) if for all s ≤ t, s, t ∈ [t_{1}, t_{2}] :

v(t) = v(s) − ∫_{s}^{t} ∇F^{ω}(x(τ), τ)dτ =

v(s) − ∫_{s}^{t} ∑_{k=1}^{N} ∇F^{k}(x(τ))Ẇ_{k}(τ)dτ =  (23)

v(s) + ∫_{s}^{t} ∑_{k=1}^{N}(v(τ) · ∇)∇F^{k}(x(τ))(W_{k}(τ) − W_{k}(s))dτ

− ∑_{k=1}^{N} ∇F^{k}(x(t))(W_{k}(t) − W_{k}(s)),

where v(t) = ẋ(t). It is well known (see [19, 31]) that the Euler-Lagrange system (22) defines a stochastic flow of diffeomorphisms of 𝕋^{d} × ℝ^{d} :

L_{s}^{ω} : (x_{0}, v_{0}) ↦ (x_{s}, v_{s}), s ∈ ℝ.  (24)

We call L_{s}^{ω} the Lagrangian flow. We shall also define the skew-product extension L̂_{s} of the Lagrangian flow. Namely, we consider the flow of non-random transformations of 𝕋^{d} × ℝ^{d} × Ω given by:

L̂_{s} : (x_{0}, v_{0}, ω) ↦ (L_{s}^{ω}(x_{0}, v_{0}), θ^{s}ω) = (x_{s}, v_{s}, θ^{s}ω), s ∈ ℝ.  (25)

In the “kicked” case the Lagrangian dynamics is given by a random Standard-like map (instead of Euler-Lagrange equation):

x_{n+1} = x_{n} + v_{n} − ∇F_{n}^{ω}(x_{n}) (mod ℤ^{d})
v_{n+1} = v_{n} − ∇F_{n}^{ω}(x_{n}).  (26)

Then the Lagrangian flow L_{n}^{ω} and its skew-product extension is defined exactly in the same way as above

L_{m}^{ω} : (x_{0}, v_{0}) ↦ (x_{m}, v_{m}), L̂_{m} : (x_{0}, v_{0}, ω) ↦ (x_{m}, v_{m}, θ^{m}ω), m ∈ ℤ.  (27)

An easy calculation shows that all minimizing sequences X are trajectories of the map (26).

Remark 3.1. In order to show that a minimizing sequence X satisfies (26) one have to differentiate A_{m,n}^{ω,b}(X) with respect to x_{i}, m ≤ i ≤ n. This may not be possible since A_{m,n}^{ω,b}(X) is not everywhere differentiable. However, it is not a problem. Indeed, one can easily show that A_{m,n}^{ω,b}(X) is differentiable if X is a minimizing sequence.

Let X_{x,n}^{ω,b} be a one-sided minimizing sequence which ends at point x ∈ 𝕋^{d} at time n ∈ ℤ. It corresponds to a trajectory {(x_{i}, v_{i}), −∞ < i ≤ n} of the random map (26). Denote by γ_{x,n}^{ω,b}(t), t ∈ (−∞, n], the piecewise linear curve on 𝕋^{d} which is formed by a particle moving from x_{i−1} to x_{i} with the velocity v_{i} :

γ_{x,n}^{ω,b}(t) = x_{i−1} + v_{i}(t − (i − 1)), i − 1 ≤ t ≤ i, −∞ < i ≤ n.  (28)

Sometimes we shall call γ_{x,n}^{ω,b}(t), t ∈ (−∞, n], a one-sided minimizer in the “kicked” case.

Finally, we define the notion of a minimizing measure for the Lagrangian flow. Denote by T 𝕋^{d} = 𝕋^{d} × ℝ^{d} the tangent bundle of 𝕋^{d} and consider a probability measure χ((dx, dv), dω) on T 𝕋^{d} × Ω. We shall say that the measure χ is an invariant measure for the random Lagrangian flow if:

1. The probability measure χ is invariant under L̂_{s} : (L̂_{s})_{∗} χ = χ, s ∈ ℝ

2. Marginal distribution of χ on Ω is given by P, i.e. χ = χ^{ω}(dx, dv)P (dω) where χ^{ω}(dx, dv) are the conditional distributions on T 𝕋^{d} under the condition of fixed ω.

As usual in the “kicked case” one has to replace s by an integer m, and L̂_{s} by L̂_{m}. Notice, that we don’t require invariant measures for the random Lagrangian flow to be “physical” in the sense of the condition 3 of Definition 1. Namely, the conditional distributions m^{ω}(dx, dv) may depend on the future behaviour of the random potential F^{ω}(·, t), t > 0. In fact, it will be exactly the case for invariant measures corresponding to global minimizers which we construct in the next section (see Remark 4.3).

4. Formulation of the main results

Consider the random Hamilton-Jacobi equation (9) with the “white force” potential (3) in a semi-infinite interval of time (−∞, t_{0}]. It turns out that for any b ∈ ℝ^{d} there exists a unique (up to a constant) “viscosity” solution. More precisely, the following theorem holds.

Theorem 1. 1. For almost all ω and all b ∈ ℝ^{d}, t_{0} ∈ ℝ there exists a unique (up to an additive constant) function ϕ_{b}^{ω}(y, t), y ∈ ℝ^{d}, t ∈ (−∞, t_{0}], such that

ϕ_{b}^{ω}(y, t) = b · y + ψ_{b}^{ω}(y, t)

where ψ_{b}^{ω} is a ℤ^{d}−periodic function and for all −∞ < s < t ≤ t_{0}

ψ_{b}^{ω}(y, t) = K_{s,t}^{ω,b}ψ_{b}^{ω}(y, s).

2. The function ψ_{b}^{ω} is Lipschitz in x : ψ_{b}^{ω}(·, t) ∈ Lip(𝕋^{d}). If x ∈ 𝕋^{d} is a point of differentiability of ψ_{b}^{ω}(x, t) then there exists a unique one-sided minimizer γ_{x,t}^{ω,b} at (x, t) and its velocity is given by the gradient of ϕ: γ̇_{x,t}^{ω,b}(t) = ∇ϕ_{b}^{ω}(x, t) = b + ∇ψ_{b}^{ω}(x, t).

3. For almost all ω and all b ∈ ℝ^{d} one has:

lim_{s→−∞} sup_{η∈C(𝕋^{d})} min_{C∈ℝ} max_{x∈𝕋^{d}}|K_{s,t}^{ω,b}η(x) − ψ_{b}^{ω}(x, t) − C| = 0.

Moreover, if x is a point of differentiability of ϕ_{b}^{ω}(x, t), so that there exists a unique one-sided minimizer at (x, t) and γ̇_{x,t}^{ω,b}(t) = ∇ϕ_{b}^{ω}(x, t), then

lim_{s→−∞} sup_{η∈C(𝕋^{d})} sup_{γ_{η,s;x,t}^{ω,b}} |γ̇_{η,s;x,t}^{ω,b}(t) − γ̇_{x,t}^{ω,b}(t)| = 0,

where γ_{η,s;x,t}^{ω,b} is a η−minimizer for A^{ω,b} on [s, t] at point x.

Statement 3 of the Theorem 1 means that for an arbitrary one-parameter family of the Cauchy problems with initial values ζ_{s}(y) = b · y + η_{s}(y) their solutions at time t converge up to an additive constant to ϕ_{b}^{ω}(y, t) as s → −∞.

Under additional assumptions which we formulate below a similar theorem holds in the “kicked” case. Denote by supp(χ) the support of the probability measure χ in the space of C^{1}−smooth potentials:

supp(χ) = {F ∈ C^{1}(𝕋^{d}) : ∀ϵ > 0, χ(O_{ϵ}(F)) > 0},

where O_{ϵ}(F) is the ϵ−ball in C^{1}(𝕋^{d}) with the centre at F. We shall consider a probability distribution χ for the kicking potentials F_{j}^{ω}(x) which satisfy one of the following assumptions.

Assumption 1. For any y ∈ 𝕋^{d} there exists a potential G_{y} belonging to supp(χ) such that G_{y} has a unique global maximum at y. For any ρ > 0 potentials G_{y} satisfy the following condition:

a(ρ) = inf_{y∈𝕋^{d}} min_{|x−y|≥ρ}(G_{y}(y) − G_{y}(x)) > 0 .  (29)

Assumption 2. Zero potential G_{0} ≡ 0 belongs to supp(χ).

Assumption 3. There exists a potential G belonging to supp(χ) with a unique global maximum.

Theorem 2. 1. Suppose probability distribution χ ∈ P(C^{1}(𝕋^{d})) satisfies Assumptions 1 or 2. Then all the statements of Theorem 1 remain true with replacement of s, t by m, n ∈ ℤ; ϕ_{b}^{ω}(y, t), ψ_{b}^{ω}(y, t), ψ_{b}^{ω}(y, s) by ϕ_{b}^{ω}(y, n−), ψ_{b}^{ω}(y, n−), ψ_{b}^{ω}(y, m−); K_{s,t}^{ω,b}, A^{ω,b} by K_{m,n}^{ω,b}, A^{ω,b}; and γ̇_{x,t}^{ω,b}(t), γ̇_{η,s;x,t}^{ω,b}(t) by γ̇_{x,n}^{ω,b}(n−), γ̇_{η,m;x,n}^{ω,b}(n−).

2. In the case b = 0 all the statements above hold under Assumption 3.

Remark 4.1. The first statement of Theorem 2 holds for all b ∈ ℝ^{d}. The second statement says that in the case b = 0 Assumption 1 can be replaced by much weaker Assumption 3. It is easy to see that Assumption 3 is equivalent to the following “mild” condition: the set of all C^{1}−smooth potentials F with a unique global maximum has a positive χ−measure.

We next formulate a very general statement which implies both Theorems 1 and 2. First we define an important notion of an ϵ−narrow place.

Definition 4. An interval of time [t_{1}, t_{2}] is called an ϵ−narrow place for an action A if there exist a subinterval [s_{1}, s_{2}] ⊂ [t_{1}, t_{2}] and sets M_{1} and M_{2} such that

1. For every minimizer γ = γ_{(x,t_{1};y,t_{2})} on [t_{1}, t_{2}] one has: γ(s_{1}) ∈ M_{1}, γ(s_{2}) ∈ M_{2} 2. If x_{1}, x_{2} ∈ M_{1} and y_{1}, y_{2} ∈ M_{2} and γ_{1} = γ_{(x_{1},s_{1};y_{1},s_{2})}, γ_{2} = γ_{(x_{2},s_{1};y_{2},s_{2})} are two

minimizers on [s_{1}, s_{2}], then

|A_{s_{1},s_{2}}(γ_{1}) − A_{s_{1},s_{2}}(γ_{2})| ≤ ϵ.

The same definition applies in the “kicked” case for minimizing sequences X, “kicked” action A and integer time intervals [m_{1}, m_{2}] ⊂ [n_{1}, n_{2}].

Consider again the Banach space C(𝕋^{d})/ℝ of equivalence classes ψ̃ = {ψ_{1} ∈ C(𝕋^{d}) : ψ_{1} = ψ + const} with the norm ∥ψ̃∥ = min_{C∈ℝ} ∥ψ − C∥_{C(𝕋^{d})}.

Proposition 1. Fix b ∈ ℝ^{d}. Suppose that for any ϵ > 0 there exists an ϵ−narrow place for an action A^{b} (A^{b}) belonging to (−∞, t_{0}] ((−∞, n_{0}]). Then

1. There exists a unique solution ψ̃(t), t ∈ (−∞, t_{0}] (ψ̃(n±), n ∈ (−∞, n_{0}]) of the factorized Hamilton-Jacobi equation (see (21)).

2. For any t ∈ (−∞, t_{0}] (n ∈ (−∞, n_{0}]) one has:

lim_{s→−∞} sup_{η̃∈C(𝕋^{d})/ℝ}∥K̃_{s,t}η̃ − ψ̃(t)∥ = 0 (in the “white force” case)

lim_{m→−∞} sup_{η̃∈C(𝕋^{d})/ℝ}∥K̃_{m,n}η̃ − ψ̃(n)∥ = 0 (in the “kicked” case)

We shall show in the next Section that Theorems 1, 2 follow easily from Proposition 1 and the following result.

Proposition 2. Consider the action A^{ω,b} in the case of “white force” potential (3) or A^{ω,b} in the “kicked” case (4). For P-almost all ω the following statements hold.

1. In the “white force” case for all b ∈ ℝ^{d}, t_{0} ∈ ℝ and all ϵ > 0 there exists an ϵ−narrow place for A^{ω,b} in (−∞, t_{0}].

2. In the “kicked case” for all b ∈ ℝ^{d}, n_{0} ∈ ℤ and all ϵ > 0 there exists an ϵ−narrow place for A^{ω,b} in (−∞, n_{0}], provided Assumptions 1 or 2 hold.

3. If Assumption 3 holds then for b = 0, all n_{0} ∈ ℤ and all ϵ > 0 there exists an ϵ−narrow place for A^{ω,0} in (−∞, n_{0}].

The following corollary is a simple consequence of Theorems 1, 2.

Corollary 1. Consider the random Burgers equation (1) in a semi-infinite interval of time (−∞, t_{0}] (or (−∞, n_{0}]) with the “white force” (3) or with the “kicked” force (4) under condition that it satisfies one of the Assumptions 1 or 2. Then for P-almost all ω the following statements hold.

1. For all b ∈ ℝ^{d} there exists a unique “viscosity” solution u_{b}^{ω}(x, t), t ∈ (−∞, t_{0}] (or u_{b}^{ω}(x, n±), n ∈ (−∞, n_{0}]) with the average velocity b.

2. For Lebesgue-almost all x

u_{b}^{ω}(x, t) = ∇ϕ_{b}^{ω}(x, t) = γ̇_{x,t}^{ω,b}(t), t ∈ (−∞, t_{0}],

where γ_{x,t}^{ω,b} is the unique one-sided minimizer at (x, t).

In the “kicked” case:

u_{b}^{ω}(x, n−) = ∇ϕ_{b}^{ω}(x, n−) = γ̇_{x,n}^{ω,b}(n−), n ∈ (−∞, n_{0}].

3. For all all b ∈ ℝ^{d}, t ∈ (∞, t_{0}] and arbitrary sequence of the Cauchy problems for the forced Burgers equation with the initial value at time s_{k}, s_{k} → −∞ as k → ∞, given by gradient-like functions w_{k}(x, s_{k}), such that ∫_{𝕋^{d}} w_{k}(x, s_{k})dx = b, the following convergence holds for Lebesgue-almost all x ∈ 𝕋^{d}:

lim_{k→∞} w_{k}(x, t) = u_{b}^{ω}(x, t),  (30)

where w_{k}(x, t) is the “viscosity” solution of the corresponding Cauchy problem. In the “kicked” case one has to replace t by n−, n ∈ ℤ in (30).

4. In the case b = 0 and “kicked” force all the statements above hold under Assumption 3.

Corollary 1 implies uniqueness of the stationary distribution for the random Burgers equation. Denote by U_{b} = {u(x) : u(x) = b + ∇ψ(x), ψ ∈ Lip(𝕋^{d})}. Obviously, U_{b} is an invariant set for the Burgers dynamics. We shall regard U_{b} as a subset of a space L^{p}(𝕋^{d}, dx), 1 ≤ p < ∞, of vector-valued functions on 𝕋^{d}. Since L^{p}(𝕋^{d}, dx) is equipped with the Borel σ−algebra B^{p}, the embedding above defines a measurable structure on U_{b}. It is easy to see that U_{b} itself is a measurable subset of L^{p}(𝕋^{d}, dx) with respect to B^{p} for all 1 ≤ p < ∞.

Lemma 1. The σ−algebras generated on U_{b} by B^{p} coincide for all 1 ≤ p < ∞.

Denote by B_{b} the corresponding σ−algebra on U_{b}. Define the following mapping U : Ω → U_{b}

U(ω) = u_{b}^{ω}(x, 0) (or U(ω) = u_{b}^{ω}(x, 0+) in the “kicked” case).  (31)

Lemma 2. The mapping U : Ω → U_{b} is measurable with respect to the σ−algebra B_{b}.

Denote by δ_{b}^{ω}(du) the atomic measure on U_{b} with the atom at u_{b}^{ω}(x, 0) (or at the u_{b}^{ω}(x, 0+) in the “kicked case”), and define a probability measure ν_{b} on P(U_{b} × Ω) by

ν_{b}(du, dω) = δ_{b}^{ω}(du)P (dω).  (32)

We shall also consider the marginal distribution of ν_{b} on U_{b}. Denote this marginal distribution by μ_{b} :

μ_{b}(du) = ∫_{Ω} δ_{b}^{ω}(du)P (dω).  (33)

Theorem 3. Consider the evolution given by the random Burgers equation (1) with the “white force” (3) or with the “kicked” force (4). In the later case assume that one of the Assumptions 1 or 2 is satisfied. Then the following statements hold.

1. For all b ∈ ℝ^{d} the probability measure ν_{b} ∈ P(U_{b} × Ω) is the unique invariant measure for the skew-product dynamics (7) on U_{b} × Ω with the marginal distribution on Ω equal to P (dω).

2. For all b ∈ ℝ^{d} the probability measure μ_{b} ∈ P(U_{b}) is the unique stationary distribution for the Markov process (8) on U_{b}.

3. In the case b = 0 and “kicked” force both statements above hold under Assumption 3.

We next discuss the properties of minimizers. It follows from Theorem 1 that for a fixed time t and for Lebesgue-almost all points x there exists a unique one-sided minimizer. Points of non-uniqueness are called shock waves, or simply shocks. Every one-sided minimizer γ_{x,t_{0}} is a trajectory of the Lagrangian flow corresponding to the Euler-Lagrange system (22), or the random map (26). It can be continued as a trajectory of Lagrangian flow for all t. However, its continuation is likely not to be a one-sided minimizer for large enough t > t_{0}. Global minimizers correspond to exactly those one-sided minimizers that can be continued as minimizers for all t. It turns out that in a very general situation there exists a unique global minimizer. We shall first formulate a theorem in the “white” force case.

Theorem 4. Suppose that a mapping

(F^{1}(x), ... , F^{N}(x)) : 𝕋^{d} → ℝ^{N}  (34)

is an embedding. Then for all b ∈ ℝ^{d} and P-almost all ω there exists a unique global minimizer x_{g}^{ω,b}(t), t ∈ ℝ, for the action A^{ω,b}.

In the “kicked” case we shall assume that the kicking potentials F_{j}^{ω}(x) are given by linear combination of the smooth non-random potentials F^{i}, 1 ≤ i ≤ M with random coefficients.

Assumption 4.

F_{j}^{ω}(x) = ∑_{i=1}^{M} ξ_{j}^{i}(ω)F^{i}(x), j ∈ ℤ,

where the random vectors ξ_{j}(ω) = {ξ_{j}^{i}(ω), 1 ≤ i ≤ M} are independent identically distributed vectors in ℝ^{M} with an absolutely continuous distribution.

Theorem 5. 1. Suppose that Assumption 4 and one of the Assumptions 1, 2 hold, and in addition the potentials F^{i}(x), 1 ≤ i ≤ M, are such that the mapping

(F^{1}(x), ..., F^{M}(x)) : 𝕋^{d} → ℝ^{M}  (35)

is one-to-one. Then for all b ∈ ℝ^{d} and P-almost all ω there exists a unique global minimizing sequence X_{g}^{ω,b}(n), n ∈ ℤ for the action A^{ω,b}.

2. In the case b = 0 one has uniqueness of a global minimizing sequence for the action A^{ω,0} for P-almost all ω, provided that Assumptions 3, 4 hold and the mapping (35) is one-to-one.

Remark 4.2.

1. In Theorems 4, 5 the full P −measure set of ω for which one has uniqueness depends on b. The stronger statement of existence of a “universal” full P −measure set of ω for which uniqueness holds for all b ∈ ℝ^{d} is probably incorrect.

2. The injectivity conditions (34), (35) are quite natural. They exclude the situation when all the potentials have an additional symmetry which results in non-uniqueness of the global minimizer. We shall discuss this in more detail in the last section.

3. It is easy to give examples of potentials for which the conditions of Theorems 4, 5 hold. Thus in the case d = 1 condition (34) holds if F^{1}(x) = sin 2πx and F^{2}(x) = cos 2πx. In the “kicked” case it is enough to consider F_{j}^{ω}(x) = ξ_{j}^{1}(ω) sin 2πx + ξ_{j}^{2}(ω) cos 2πx . Obviously Assumptions 2, 4 and condition (35) hold if (ξ_{j}^{1}(ω) , ξ_{j}^{2}(ω)) have positive density inside a disc D_{r} = {(ξ^{1})^{2} + (ξ^{2})^{2} ≤ r} , r > 0. It is easy to see that Assumption 1 is satisfied if (ξ_{j}^{1}(ω) , ξ_{j}^{2}(ω)) have positive density inside an annulus A_{r_{1},r_{2}} = {r_{1} ≤ (ξ^{1})^{2} + (ξ^{2})^{2} ≤ r_{2}} , r_{2} > r_{1} > 0.

Denote by x_{g}^{ω,b} the position of the unique global minimizer or minimizing sequence at time t_{0} = 0, or n_{0} = 0. For fixed t > 0 or n > 0 consider all one-sided minimizers γ_{x,t}^{ω,b} or one-sided minimizing sequences X_{x,n}^{ω,b}, x ∈ 𝕋^{d}. Denote by E^{ω,b}(t) the set of points at which one-sided minimizers or minimizing sequences cross 𝕋^{d} at t = 0: E^{ω,b}(t) = {x_{0} = γ_{x,t}^{ω,b}(0)} in the “white” case, or E^{ω,b}(t) = {x_{0} = X_{x,n}^{ω,b}(0)} in the “kicked” case. Notice, that x_{g}^{ω,b} ∈ E^{ω,b}(t). The following statement follows immediately from the uniqueness of the global minimizer.

Corollary 2. If there exists a unique global minimizer or minimizing sequence then diam(E^{ω,b}(t)) → 0 as t → ∞.

As we have mentioned above the unique global minimizer or global minimizing sequence gives rise to an invariant measure for the Lagrangian flow. To define this measure consider the velocity v_{g}^{ω,b} of the global minimizer at time t_{0} = 0 : v_{g}^{ω,b} = ẋ_{g}^{ω,b}(0): In “kicked” case v_{g}^{ω,b} = v_{0}, where {(x_{n}, v_{n}), n ∈ ℤ} is the trajectory of the random map (26) corresponding to the global minimizing sequence X_{g}^{ω,b}(n), n ∈ ℤ. It is easy to see that v_{0} is the velocity at time n = 0−. Define a probability measure κ ∈ P(𝕋^{d}×ℝ^{d}×Ω) by

κ^{b}(dx, dv, dω) = δ_{x_{g}^{ω,b},v_{g}^{ω,b}}(dx, dv)P (dω),  (36)

where δ_{x_{g}^{ω,b},v_{g}^{ω,b}}(dx, dv) is an atomic measure in P(𝕋^{d} ×ℝ^{d}) with the atom at (x_{g}^{ω,b}, v_{g}^{ω,b}).

Corollary 3. Suppose all the assumptions of Theorems 4, 5 are satisfied. Then for all b ∈ ℝ^{d} the probability measure κ^{b}(dx, dv, dω) is an ergodic invariant measure for the skew-product extension L̂_{s} (or L̂_{m} in the “kicked” case) of the Lagrangian flow.

Remark 4.3. As we have already mentioned above the minimizing measures κ^{b}(dx, dv, dω) although invariant under skew-product extension of the Lagrangian flow, does not satisfy Condition 3 of Definition 1. Indeed, the random variables x_{g}^{ω,b}, v_{g}^{ω,b} are measurable only with respect to the full σ−algebra F_{−∞}^{∞} and are not measurable with respect to the “past” σ−algebra F_{−∞}^{0}.

5. Uniqueness of Solutions for Random Hamilton-Jacobi equation

We shall first prove Proposition 1, then derive Theorems 1, 2 and Corollary 1 from Propositions 1, 2. Finally we shall prove Proposition 2.

Except for Proposition 2 the proofs in the “white” and the “kicked” cases are the same. So we only consider the “white force” case. Since for all the statements with the exception of Proposition 2 the dependence on b is not essential, we fix b and omit it from most of our notations.

For the proof of Proposition 1 we need the following three lemmas. Let Ω_{1} ⊂ Ω be the set of ω such that the realizations of the Wiener processes W_{k}(t), 1 ≤ k ≤ N, are continuous functions of t. Obviously P (Ω_{1}) = 1.

Lemma 3. For all ω ∈ Ω_{1} and for all −∞ < s < t < ∞, the operator K_{s,t}^{ω} is a weak contraction in the uniform topology, that is for every pair ψ_{1}, ψ_{2} ∈ C(𝕋^{d}) we have:

||K_{s,t}^{ω}ψ_{1} − K_{s,t}^{ω}ψ_{2}||_{C} ≤ ||ψ_{1} − ψ_{2}||_{C}  (37)

Proof. Recall that

K_{s,t}^{ω}ψ_{1}(x) = inf_{γ∈AC(s;x,t)} (ψ_{1}(γ(s)) + A_{s,t}^{ω}(γ)) .

Let γ = γ_{(ψ_{1},s;x,t)} be a ψ_{1}- minimizer, i.e. a curve such that infimum in the above equation is realized. We have:

K_{s,t}^{ω}ψ_{1}(x) = ψ_{1}(γ(s)) + A_{s,t}^{ω}(γ),

and

K_{s,t}^{ω}ψ_{2}(x) ≤ ψ_{2}(γ(s)) + A_{s,t}^{ω}(γ).

Subtracting we get

K_{s,t}^{ω}ψ_{2}(x) − K_{s,t}^{ω}ψ_{1}(x) ≤ ψ_{2}(γ(s)) − ψ_{1}(γ(s)) ≤ ||ψ_{1} − ψ_{2}||_{C}.

Interchanging ψ_{1} and ψ_{2} we obtain

|K_{s,t}^{ω}ψ_{2}(x) − K_{s,t}^{ω}ψ_{1}(x)| ≤ ||ψ_{1} − ψ_{2}||_{C},

which implies (37).

Lemma 4. (Semigroup Property) For all ω ∈ Ω_{1} and for all −∞ < s < τ < t < ∞ one has:

K_{s,t}^{ω} = K_{τ,t}^{ω} ∘ K_{s,τ}^{ω}.  (38)

Moreover if γ = γ_{(ψ,s;x,t)}^{ω} is a ψ−minimizer and ψ_{τ} = K_{s,τ}^{ω}ψ then γ|_{[τ,t]} is a ψ_{τ}−minimizer on [τ, t].

Proof. By definition we have:

K_{τ,t}^{ω} ∘ K_{s,τ}^{ω}ψ(x) = inf_{σ∈AC(τ;x,t)} (K_{s,τ}^{ω}ψ(σ(τ)) + A_{τ,t}^{ω}(σ))
= inf_{β∈AC(s;σ(τ),τ),σ∈AC(τ;x,t)} (ψ(β(s)) + A_{s,τ}^{ω}(β) + A_{τ,t}^{ω}(σ))
= inf_{γ∈AC(s;x,t)} (ψ(γ(s)) + A_{s,t}^{ω}(γ))
= K_{s,t}^{ω}ψ(x).

To prove the second assertion suppose that there exist σ_{1} such that

A_{τ,t}^{ω}(σ_{1}) + ψ_{τ}(σ_{1}(τ)) < A_{τ,t}^{ω}(γ) + ψ_{τ}(γ(τ)).

Denote by σ_{2} a minimizer γ_{(ψ,s;σ_{1}(τ),τ)}^{ω} and by σ : [s, t] → 𝕋^{d} the curve which coincides with σ_{1} on [τ, t] and with σ_{2} on [s, τ]. We have:

ψ_{τ}(σ_{1}(τ)) = A_{s,τ}^{ω}(σ_{2}) + ψ(σ_{2}(s)),
ψ_{τ}(γ(τ)) = A_{s,τ}^{ω}(γ) + ψ(γ(s)).

Hence,

A_{s,t}^{ω}(σ) + ψ(σ(s)) = A_{s,τ}^{ω}(σ_{2}) + A_{τ,t}^{ω}(σ_{1}) + ψ(σ_{2}(s)) < A_{s,t}^{ω}(γ_{)} + ψ(γ(s)),

which is a contradiction.

Remark 5.1. It follows immediately from Lemmas 3, 4 that

||K̃_{s,t}^{ω}ψ̃_{1} − K̃_{s,t}^{ω}ψ̃_{2}|| ≤ ||ψ̃_{1} − ψ̃_{2}||  (39)

and

K̃_{s,t}^{ω} = K̃_{τ,t}^{ω} ∘ K̃_{s,τ}^{ω}.  (40)

Lemma 5. (Main Lemma) If [t_{1}, t_{2}] is an ϵ-narrow place then for every ψ_{1}, ψ_{2} ∈ C(𝕋^{d}) there exist a constant c such that

|K_{t_{1},t_{2}}^{ω}ψ_{1}(x) + c − K_{t_{1},t_{2}}^{ω}ψ_{2}(x)| ≤ 3ϵ  (41)

Proof. Let [s_{1}, s_{2}] ⊂ [t_{1}, t_{2}] be the subinterval and M_{1}, M_{2} subsets of 𝕋^{d} from the Definition 4. Suppose γ is an arbitrary ψ−minimizer on [t_{1}, t_{2}]. Then for all p ∈ M_{1} we have:

ψ′(p) ≥ ψ′(γ(s_{1})) − ϵ,  (42)

where ψ′ = K_{t_{1},s_{1}}^{ω}ψ. To prove (42) consider α = γ_{ψ,t_{1};p,s_{1}}, β = γ_{(p,s_{1};γ(s_{2}),s_{2})} and

γ̄(τ) = \begin{cases}α if τ ∈ [t_{1}, s_{1}] \\ β if τ ∈ [s_{1}, s_{2}] \\ γ if τ ∈ [s_{2}, t_{2}] \end{cases}

If (42) does not hold we get:

ψ(γ̄(t_{1})) + A_{t_{1},t_{2}}^{ω}(γ̄) = ψ′(p) + A_{s_{1},t_{2}}^{ω}(γ̄)
< ψ′(γ(s_{1})) + A_{s_{1},s_{2}}^{ω}(β) + A_{s_{2},t_{2}}^{ω}(γ) − ϵ
≤ ψ′(γ(s_{1})) + A_{s_{1},s_{2}}^{ω}(γ) + ϵ + A_{s_{2},t_{2}}^{ω}(γ) − ϵ
= ψ′(γ(s_{1})) + A_{s_{1},t_{2}}^{ω}(γ) = ψ(γ(t_{1})) + A_{t_{1},t_{2}}^{ω}(γ),

which contradicts to the fact that γ is a ψ−minimizer on [t_{1}, t_{2}]. For any ψ ∈ C(𝕋^{d}) denote by M_{ψ} the set of points p = γ_{(ψ,t_{1};x,t_{2})}^{ω}(s_{1}) for some x ∈ 𝕋^{d}. Clearly M_{ψ} is a subset of M_{1}. Using (42) we get

|ψ′(p, s_{1}) − ψ′(q, s_{1})| ≤ ϵ,  (43)

for all p, q in M_{ψ}. Let ψ_{1}, ψ_{2} ∈ C(𝕋^{d}). The estimate (43) implies that there exist a constant c such that for all p ∈ M_{ψ_{1}} and q ∈ M_{ψ_{2}} we have

|ψ′_{1}(p) + c − ψ′_{2}(q)| ≤ 2ϵ.  (44)

Finally, let γ_{1}, γ_{2} be ψ_{1}, ψ_{2}−minimizers at x ∈ 𝕋^{d} : γ_{1} = γ_{(ψ_{1},t_{1};x,t_{2})}^{ω}, γ_{2} = γ_{(ψ_{2},t_{1};x,t_{2})}^{ω}. Denote p_{1} = γ_{1}(s_{1}), p_{2} = γ_{1}(s_{2}), q_{1} = γ_{2}(s_{1}), q_{2} = γ_{2}(s_{2}). Also denote by δ the minimizer γ_{(p_{1},s_{1};q_{2},s_{2})}^{ω}. Then we have:

K_{t_{1},t_{2}}^{ω}ψ_{1}(x) ≤ ψ′_{1}(p_{1}) + A_{s_{1},s_{2}}^{ω}(δ) + A_{s_{2},t_{2}}^{ω}(γ_{2}),
K_{t_{1},t_{2}}^{ω}ψ_{2}(x) = ψ′_{2}(q_{1}) + A_{s_{1},s_{2}}^{ω}(γ_{2}) + A_{s_{2},t_{2}}^{ω}(γ_{2}).

Hence

K_{t_{1},t_{2}}^{ω}ψ_{1}(x) + c − K_{t_{1},t_{2}}^{ω}ψ_{2}(x) ≤ ψ′_{1}(p_{1}) + c − ψ′_{2}(q_{1}) + A_{s_{1},s_{2}}^{ω}(δ) − A_{s_{1},s_{2}}^{ω}(γ_{2})
≤ 2ϵ + ϵ ≤ 3ϵ.

Interchanging ψ_{1} and ψ_{2} we get (41).

An easy but useful consequence of the above lemmas is the following corollary.

Corollary 4. If the interval [s, t] contains an ϵ−narrow place then for every ψ̃_{1}, ψ̃_{2} ∈ C(𝕋^{d})/ℝ we have ||K̃_{s,t}^{ω}ψ̃_{1} − K̃_{s,t}^{ω}ψ̃_{2}|| ≤ 3ϵ.

Proof. Let [t_{1}, t_{2}] be an ϵ−narrow place which is contained inside the interval [s, t]. Using Remark 5.1 we can decompose the operator K̃_{s,t} into three parts:

K̃_{s,t}^{ω} = K̃_{t_{2},t}^{ω} ∘ K̃_{t_{1},t_{2}}^{ω} ∘ K̂_{s,t_{1}}^{ω}.

It follows from (39) that the last part K̃_{t_{2},t}^{ω} does not increase the distance. Now it is enough to notice that (41) implies ||K̃_{t_{1},t_{2}}^{ω}η̃_{1} − K̃_{t_{1},t_{2}}^{ω}η̃_{2}|| < 3ϵ for all η̃_{1}, η̃_{2} ∈ C(𝕋^{d})/ℝ.

Proof of Proposition 1 . We show that for all η̃ ∈ C(𝕋^{d})/ℝ and all t ∈ ℝ fixed K̃_{s,t}^{ω}η̃ satisfies the Cauchy property, i.e for all δ > 0 there exists S < t such that for all s_{1}, s_{2} < S :

||K̃_{s_{1},t}^{ω}η̃ − K̃_{s_{2},t}^{ω}η̃|| ≤ δ.  (45)

Indeed, let [t_{1}, t_{2}] be an \frac{δ}{3}−narrow place inside (−∞, t]. Put S = t_{1}. Then (45) follows immediately from Corollary 4 and the semigroup property 40.

Since C(𝕋^{d})/ℝ is a Banach space K̃_{s,t}^{ω}ψ̃ converges as s → −∞. Denote the limit by ψ̃^{ω}(t). It follows from Corollary 4 that the limit does not depend on the initial η. In fact, Corollary 4 implies the following statement:

lim_{s→−∞} sup_{η̃∈C(𝕋^{d})/ℝ}∥K̃_{s,t}^{ω}η̃ − ψ̃^{ω}(t)∥ = 0 ,

which proves the second part of Proposition 1. We next prove that ψ̃^{ω}(t), t ∈ ℝ is a solution of the factorized Hamilton - Jacobi equation. Take arbitrary s < τ < t and consider an identity:

K̃_{s,t}^{ω}η̃ = K̃_{τ,t}^{ω} ∘ K̃_{s,τ}^{ω} η̃.

Taking the limit as s → −∞ we obtain

ψ̃^{ω}(t) = K̃_{τ,t}^{ω}ψ̃^{ω}(τ).

To complete the proof we just notice that Corollary 4 obviously implies the uniqueness of the solution ψ̃^{ω}(t).

For the proof of Theorem 1 we shall need the following lemmas.

Lemma 6. For all ω ∈ Ω_{1} and all −∞ < s < t < ∞ there exists a constant C(ω, s, t) such that:

sup_{x_{1}, x_{2}∈𝕋^{d}} sup_{γ_{(x_{1},s;x_{2},t)}^{ω}} max_{τ∈[s,t]} |γ̇_{(x,s;y,t)}^{ω}(τ)| ≤ C(ω, s, t) ,  (46)

where |v| = ∥v∥ = \sqrt{v · v}, v ∈ ℝ^{d}.

Proof. From the formula (23) it follows that there exist positive constants K_{1}(s, t, ω), K_{2}(s, t, ω) such that

|v(τ)| ≤ |v(s)| + K_{1}(s, t, ω) + K_{2}(s, t, ω) ∫_{s}^{τ} |v(u)|du.

Hence, using the Gronwall inequality we have:

|v(τ)| ≤ (|v(s)| + K_{1}(s, t, ω))e^{K_{2}(s,t,ω)(τ−s)}.  (47)

The estimate (47) implies that for any v_{0} > 0 there exists V^{ω}(v_{0}) such that if |v(τ_{0})| ≥ V^{ω}(v_{0}) for some τ_{0} ∈ [s, t] then |v(τ)| ≥ v_{0} for all τ ∈ [s, t]. Denote by Γ(v_{0}) = {γ ∈ C^{1}[s, t] : |γ̇(τ)| ≥ v_{0} for all τ ∈ [s, t]}. Let

A^{ω}(v_{0}) = inf_{γ∈Γ(v_{0})} A_{s,t}^{ω}(γ).

It follows immediately from (16) that A^{ω}(v_{0}) → ∞ as v_{0} → ∞. Denote by δ_{(x_{1},s;x_{2},t)} a curve with a constant velocity v ≡ δ̇_{(x_{1},s;x_{2},t)}(τ), τ ∈ [s, t] connecting points x_{1} and x_{2}. It is easy to see that for all x_{1}, x_{2} one can always choose δ_{(x_{1},s;x_{2},t)} in such a way that |v| ≤ \frac{1}{2}(t − s)\sqrt{d}. Let Δ be the set of all curves δ_{(x_{1},s;x_{2},t)} for different x_{1}, x_{2} ∈ 𝕋^{d} satisfying the condition |v| ≤ \frac{1}{2}(t − s)\sqrt{d}. Denote by Ā^{ω} the supremum of actions of the curves γ̄ ∈ Δ :

Ā^{ω} = sup_{δ∈Δ} A_{s,t}^{ω}(δ).

Take now v_{0} so big that A^{ω}(v_{0}) > Ā^{ω}. Then, obviously,

|γ̇_{(x,s;y,t)}^{ω}(τ)| ≤ V (v_{0}).

Remark 5.2. In the “kicked” case one can give a more explicit estimate for velocities of minimizers. Suppose X = {x_{i} ∈ 𝕋^{d}, m ≤ i ≤ n} is a minimizing sequence for the action A_{m,n}^{ω,b}, where b = (b^{j}, 1 ≤ j ≤ d). Let v_{i}, m + 1 ≤ i ≤ n , be the velocities corresponding to the minimizer X : γ̇_{x,n}^{ω,b}(i−) = v_{i} (see (26) , (28) ). It is easy to see that for all m + 1 ≤ i ≤ n , 1 ≤ j ≤ d one has |v_{i}^{j} − b^{j}| ≤ \frac{1}{2} . Hence,

|v_{i}| ≤ |b| + \frac{\sqrt{d}}{2}, m + 1 ≤ i ≤ n .  (48)

Next lemma is a very general existence statement which follows easily from estimates (46), (48).

Lemma 7. Consider the action A^{ω,b} corresponding to the “white force” potential (3) or A^{ω,b} in the case of “kicked” potential (4). In both cases there exists a subset of Ω of full P −measure such that the following statements hold.

1. For all b ∈ ℝ^{d}, x ∈ 𝕋^{d}, t ∈ ℝ, or n ∈ ℤ in the “kicked” case, there exists at least one one-sided minimizer γ_{x,t}^{ω,b}(τ), τ ∈ (−∞, t], which ends at (x, t), or, in the “kicked” case, there exists at least one one-sided minimizing sequence X_{x,n}^{ω,b}(m), m ∈ (−∞, n], which ends at (x, n).

2. For all b ∈ ℝ^{d} there exists at least one global minimizer γ^{ω,b}(τ), τ ∈ ℝ, or, in the “kicked” case, there exists at least one global minimizing sequence X^{ω,b}(m), m ∈ ℤ.

Proof. The proof is based on a general compactness argument. For an arbitrary (x, t) consider a sequence of finite-time minimizers γ_{n}(τ) = γ_{x_{n},−n;x,t}(τ), τ ∈ [−n, t]. Denote v_{n} = γ̇_{n}(t). It follows from (46), (48) that |v_{n}| are uniformly bounded. Hence there exists a convergent subsequence v_{n_{k}} → v as k → ∞. Denote by γ_{v}(τ), τ ∈ (−∞, t] a solution of Euler-Lagrange equation such that γ_{v}(t) = x, γ̇_{v}(t) = v . Since the set of all minimizers is closed one can easily show that γ_{v} is a one-sided minimizer at (x, t). To construct a global minimizer take again a sequence of finite-time minimizers γ̄_{n}(τ) = γ_{x_{n},−n;x_{n},n}(τ). Denote y_{n} = γ̄_{n}(0), v̄_{n} = ˙ γ̄_{n}(0). Again there exists a convergent subsequence y_{n_{k}} → y, v̄_{n_{k}} → v̄ as k → ∞. It is easy to see that a solution of Euler-Lagrange equation γ_{0}(τ), τ ∈ ℝ with the initial data γ_{0}(0) = y, γ̇_{0}(0) = v̄ is a global minimizer.

In the following two lemmas we demonstrate close connection between the notions one-sided minimizer and ψ−minimizer where ψ is a solution to the Hamilton-Jacobi equation. Both lemmas are formulated for the “white force” case. The generalization for the “kicked” case is straightforward.

Lemma 8. Let ϕ(x, t) = b · y + ψ(x, t), t ∈ (−∞, t_{0}] be an arbitrary solution to the Hamilton-Jacobi equation in (x ∈ 𝕋^{d}, t ∈ [−∞, t_{0}]) and γ̄(τ), τ ∈ [t_{1}, t_{0}] be a ψ−minimizer on [t_{1}, t_{0}]. Then γ̄ can be continued to a one-sided minimizer. Namely, there exists a one-sided minimizer γ(τ), τ ∈ (−∞, t_{0}] such that γ(τ) = γ̄(τ), τ ∈ [t_{1}, t_{0}]. In addition γ(τ), τ ∈ [t, t_{0}] is a ψ−minimizer on [t, t_{0}] for any t < t_{0}.

Proof. Let γ̄(τ), τ ∈ [t_{1}, t_{0}] be a ψ−minimizer on [t_{1}, t_{0}]. Denote by γ̄_{1}(τ), τ ∈ [t_{1}−1, t_{1}] a ψ−minimizer on [t_{1} − 1, t_{1}] which starts at a point (γ̄(t_{1}), t_{1}) and by γ_{1}(τ), τ ∈ [t_{1} − 1, t_{0}] the curve which is the union of γ̄ and γ̄_{1}. It is easy to see that γ_{1} is a ψ−minimizer on [t_{1} − 1, t_{0}]. Indeed for any other curve σ(τ), τ ∈ [t_{1} − 1, t_{0}] we have

A_{t_{1},t_{0}}(γ̄) + ψ(γ̄(t_{1}), t_{1}) = A_{t_{1}−1,t_{0}}(γ) + ψ(γ(t_{1} − 1), t_{1} − 1)

and

A_{t_{1},t_{0}}(σ) + ψ(σ(t_{1}), t_{1}) ≤ A_{t_{1}−1,t_{0}}(σ) + ψ(σ(t_{1} − 1), t_{1} − 1).

Since

A_{t_{1},t_{0}}(γ̄) + ψ(γ̄(t_{1}), t_{1}) ≤ A_{t_{1},t_{0}}(σ) + ψ(σ(t_{1}), t_{1})

we conclude that

A_{t_{1}−1,t_{0}}(γ) + ψ(γ(t_{1} − 1), t_{1} − 1) ≤ A_{t_{1}−1,t_{0}}(σ) + ψ(σ(t_{1} − 1), t_{1} − 1).

In the same way we can construct a ψ−minimizer γ_{2}(τ), τ ∈ [t_{1} − 2, t_{0}] which is a continuation of γ_{1} and so on. By induction we get a curve γ(τ), τ ∈ (−∞, t_{0}] which is a ψ−minimizer on [t, t_{0}] for any t < t_{0} and coincides with γ̄(τ) for all τ ∈ [t_{1}, t_{0}]. It is also obvious that γ is a one-sided minimizer.

Lemma 9. Suppose that for any t_{0} ∈ ℝ and any ϵ > 0 there exists an ϵ−narrow place for an action A^{ω,b} belonging to (−∞, t_{0}]. Let ϕ_{b}^{ω}(x, t) = b·y +ψ_{b}^{ω}(x, t), t ∈ (−∞, t_{0}] be a solution to the Hamilton-Jacobi equation in (x ∈ 𝕋^{d}, t ∈ [−∞, t_{0}]). Then the following statements hold.

1. Any one-sided minimizer γ_{0}(τ) = γ_{x,t_{0}}^{ω,b}(τ), τ ∈ (−∞, t_{0}], is also a ψ−minimizer for ψ = ψ_{b}^{ω}(x, t) and all t ≤ t_{0}.

2. Suppose two one-sided minimizers γ_{x_{1},t_{1}}^{ω,b}(τ), γ_{x_{2},t_{2}}^{ω,b}(τ) intersect, i.e. there exists τ_{0} ≤ min(t_{1}, t_{2}) such that γ_{x_{1},t_{1}}^{ω,b}(τ_{0}) = γ_{x_{2},t_{2}}^{ω,b}(τ_{0}). Then either x_{1} = x_{2} and t_{1} = t_{2}, or γ_{x_{1},t_{1}}^{ω,b}(τ) = γ_{x_{2},t_{2}}^{ω,b}(τ) for all τ ≤ min(t_{1}, t_{2}).

Proof. We start with the proof of the first statement. Suppose γ_{0} is not a ψ−minimizer for some t̄< t_{0}. Then for some ϵ > 0,

A_{t̄,t_{0}}(σ̄) + ψ(σ̄(t̄), t̄) + ϵ = A_{t̄,t_{0}}(γ_{0}) + ψ(γ(t̄), t̄),

where σ̄ is a ψ−minimizer on [t̄, t_{0}] at point x. Let [t_{1}, t_{2}] be a \frac{ϵ}{8}− narrow place, with t_{2} < t̄, and [s_{1}, s_{2}] ⊂ [t_{1}, t_{2}] the corresponding subinterval (see Definition 4). Denote by σ a continuation of σ̄ to a ψ−minimizer on [t_{1}, t_{0}]. Obviously, for any τ in [t_{1}, t̄] we have

A_{τ,t_{0}}(γ_{0}) = A_{τ,t̄}(γ_{0}) + A_{t̄,t_{0}}(γ_{0})
≥ A_{t̄,t_{0}}(γ_{0}) + ψ(γ_{0}(t̄), t̄) − ψ(γ_{0}(τ), τ)
= A_{t̄,t_{0}}(σ) + ψ(σ(t̄), t̄) + ϵ − ψ(γ_{0}(τ), τ)
= A_{τ,t_{0}}(σ) + ψ(σ(τ), τ) + ϵ − ψ(γ_{0}(τ), τ).

In particular, we get for τ = s_{2} :

A_{s_{2},t_{0}}(σ) + ψ(σ(s_{2}), s_{2}) + ϵ ≤ A_{s_{2},t_{0}}(γ_{0}) + ψ(γ_{0}(s_{2}), s_{2}).  (49)

On the other hand,

ψ(γ_{0}(s_{2}), s_{2}) − ψ(σ(s_{2}), s_{2}) ≤ \frac{ϵ}{4};  (50)

Indeed, since σ is a ψ−minimizer, σ(s_{1}) ∈ M_{1}, σ(s_{2}) ∈ M_{2}. Similarly, γ_{0}(s_{1}) ∈ M_{1}, γ_{0}(s_{2}) ∈ M_{2}. Consider now

γ = \begin{cases}σ  for τ ∈ [t_{1}, s_{1}] \\ γ_{σ(s_{1}),s_{1};γ_{0}(s_{2}),s_{2}}  for τ ∈ [s_{1}, s_{2}] . \end{cases}

Then

ψ(σ(s_{2}), s_{2}) = A_{t_{1},s_{1}}(σ) + A_{s_{1},s_{2}}(σ) + ψ(σ(t_{1}), t_{1}),

and

ψ(γ_{0}(s_{2}), s_{2}) ≤ A_{t_{1},s_{1}}(σ) + A_{s_{1},s_{2}}(γ) + ψ(σ(t_{1}), t_{1}).

Subtracting and using

|A_{s_{1},s_{2}}(σ)|, |A_{s_{1},s_{2}}(γ)| ≤ \frac{ϵ}{8}

one gets (50). Combining (49) and (50) we have

A_{s_{2},t_{0}}(γ_{0}) ≥ A_{s_{2},t_{0}}(σ) + \frac{3ϵ}{4}.

Finally, using a minimizing curve γ_{γ_{0}(s_{1}),s_{1};σ(s_{2}),s_{2}} we construct a curve

γ̃ = \begin{cases}γ_{γ_{0}(s_{1}),s_{1};σ(s_{2}),s_{2}}  for τ ∈ [s_{1}, s_{2}] \\ σ  for τ ∈ [s_{2}, t_{0}] . \end{cases}

Since |A_{s_{1},s_{2}}(γ_{0})|, |A_{s_{1},s_{2}}(α)| ≤ \frac{ϵ}{8}, we obtain

A_{s_{1},t_{0}}(γ_{0}) = A_{s_{1},s_{2}}(γ_{0}) + A_{s_{2},t_{0}}(γ_{0}) ≥ A_{s_{1},s_{2}}(α) + As_{2}, t_{0}(σ) + \frac{ϵ}{2}
= A_{s_{1},t_{0}}(γ̃) + \frac{ϵ}{2} ,

which is contradiction with the fact that γ_{0} is a one-sided minimizer.

To prove the second statement suppose that γ_{1}(tau) = γ_{x_{1},t_{1}}^{ω,b}(τ), γ_{2}(τ) = γ_{x_{2},t_{2}}^{ω,b}(τ) have a “proper” intersection. Assume without loss of generality that t_{1} ≥ t_{2}. Then there exists t < t_{1} such that γ_{1}(t) = γ_{2}(t). It was shown above that for any t̄ < t the curves γ_{1}, γ_{2} are ψ−minimizers on [t̄, t_{1}] and [t̄, t_{1}] respectively. This implies that a curve

γ̃_{3}(τ) = \begin{cases}γ_{2}(τ)  for τ ∈ [t̄, t] \\ γ_{1}(τ)  for τ ∈ [t, t_{1}] . \end{cases}

is also a ψ−minimizers on [t̄, t_{1}]. However γ_{3}(τ) is not a solution of the Euler-Lagrange equation. This contradiction finishes the proof.

Remark 5.3. The classical result in the non-random case states that two minimizers cannot intersect twice. It is interesting to compare this fact with the second statement of Lemma 9 which implies that in the presence of ϵ−narrow places two one-sided minimizers cannot have even one “proper” intersection. An intersection is possible only when both one-sided minimizers γ_{1}, γ_{2}start at the same point (x, t). It follows from the lemma above that both γ_{1} and γ_{2} cannot be continued as one-sided minimizers beyond time t. Points (x, t) of non-uniqueness of one-sided minimizers are called shocks. As we show below they correspond to singularities of ψ_{b}^{ω} (see Theorems 1, 2).

The statement of the following lemma is a well-known general fact in the case when a force depends smoothly on time. We show that the same statement holds in the “white force” setting.

Lemma 10. Let γ_{0}(τ), τ ∈ [s, t], be an arbitrary solution to the Euler-Lagrange system of equations (22) and γ(τ, ϵ), τ ∈ [s, t], be a C^{1}−smooth variation of γ_{0} with the fixed end point at time s :

γ(τ, 0) = γ_{0}(τ), τ ∈ [s, t] ; γ(s, ϵ) = γ_{0}(s).

Then

\frac{∂A_{s,t}^{ω}(γ(·, ϵ)}{∂ϵ}|_{ϵ=0} = (γ̇_{0}(t) − b) · \frac{∂γ(t, ϵ)}{∂ϵ}|_{ϵ=0} .  (51)

Proof. Define η(τ) = \frac{∂γ(τ,ϵ)}{∂ϵ}|_{ϵ=0}. From the formula (16) for the action we have:

\frac{∂A_{s,t}^{ω}(γ(·, ϵ)}{∂ϵ}|_{ϵ=0} = ∫_{s}^{t}(γ̇_{0}(τ) − b) · η̇(τ)dτ

+ ∫_{s}^{t} ∑_{k=1}^{N} [(γ̇_{0}(τ) · ∇)(∇F^{k}(γ_{0}(τ)) · η(τ))(W_{k}(τ) − W_{k}(s))] dτ

+ ∫_{s}^{t} ∑_{k=1}^{N} [(∇F^{k}(γ_{0}(τ)) · η̇(τ))(W_{k}(τ) − W_{k}(s))] dτ  (52)

− ∑_{k=1}^{N}(∇F^{k}(γ_{0}(t)) · η(t))(W_{k}(t) − W_{k}(s)) .

Using (23) for v(τ) = γ̇_{0}(τ) we obtain:

∫_{s}^{t}(γ̇_{0}(τ) − b) · η̇(τ)dτ = ∫_{s}^{t} γ̇_{0}(s) · η̇(τ)dτ

+ ∫_{s}^{t} (∫_{s}^{τ} ∑_{k=1}^{N}(γ̇_{0}(ρ) · ∇)∇F^{k}(γ_{0}(ρ))(W_{k}(ρ) − W_{k}(s))dρ) · η̇(τ)dτ

− ∫_{s}^{t} ∑_{k=1}^{N}(∇F^{k}(γ_{0}(τ)) · η̇(τ))(W_{k}(τ) − W_{k}(s))dτ

− ∫_{s}^{t} b · η̇(τ)dτ .

Integration the first and the last integrals and using integration by parts in the second we get:

∫_{s}^{t}(γ̇_{0}(τ) − b) · η̇(τ)dτ = γ̇_{0}(s) · η(t)

− ∫_{s}^{t} ∑_{k=1}^{N} [(γ̇_{0}(τ) · ∇)(∇F^{k}(γ_{0}(τ)) · η(τ))(W_{k}(τ) − W_{k}(s))] dτ

+ (∫_{s}^{t} ∑_{k=1}^{N}(γ̇_{0}(ρ) · ∇)∇F^{k}(γ_{0}(ρ))(W_{k}(ρ) − W_{k}(s))dρ) · η̇(t)  (53)

− ∫_{s}^{t} ∑_{k=1}^{N}(∇F^{k}(γ_{0}(τ)) · η̇(τ))(W_{k}(τ) − W_{k}(s))dτ − b · η(t) .

Combining (52), (53) and using again (23) we finally get:

\frac{∂A_{s,t}^{ω}(γ(·, ϵ)}{∂ϵ}|_{ϵ=0} = η(t) · (v(s)

+ ∫_{s}^{t} ∑_{k=1}^{N}(γ̇_{0}(ρ) · ∇)∇F^{k}(γ_{0}(ρ))(W_{k}(ρ) − W_{k}(s))dρ

− ∑_{k=1}^{N}(∇F^{k}(γ_{0}(t))(W_{k}(t) − W_{k}(s)) − b) =
= η(t) · (γ̇_{0}(t) − b) .

In the proof of the following lemma we shall use the following notation. For all −∞ < s < t < ∞ denote by

∥F ∥_{s,t}^{ω} = max_{s≤τ≤t} ∑_{k=1}^{N} ∥F^{k}∥_{C^{2}(𝕋^{d})}|W_{k}(τ) − W_{k}(s)| .  (54)

Lemma 11. For all ω in Ω_{1} and every −∞ < s < t < ∞ there exists a constant c(ω, s, t) ≥ 0 such that for every continuous function ψ the function given by K_{s,t}^{ω}ψ : 𝕋^{d} → ℝ is Lipschitz continuous and its Lipschitz constant is less or equal than c(ω, s, t). Moreover there exists a constant c(ω, t) ≥ 0 such that, K_{s,t,ω}ψ has Lipschitz constant less or equal than c(ω, t) for every continuous ψ and all s ≤ t − 1.

Proof. Take arbitrary x, y ∈ 𝕋^{d}. Let γ_{x} be a minimizer defined by γ_{x}(τ) = γ_{(ψ,s;x,t)}^{ω}(τ). Denote by γ a curve such that its lifting into universal cover γ̃ is equal to

γ̃(τ) = γ̃_{x}(τ) + \frac{(ỹ − x̃)(τ − s)}{t − s}, τ ∈ [s, t],  (55)

where γ̃_{x}, x̃, ỹ are liftings of γ_{x}, x and y respectively. Obviously, γ(s) = γ_{x}(s), γ(t) = y. It follows from (55) that for all τ ∈ [s, t]

dist(γ_{x}(τ), γ(τ)) ≤ dist(y, x), |γ̇_{x}(τ) − γ̇(τ)| ≤ \frac{dist(y, x)}{t − s}.  (56)

Since

K_{s,t}^{ω}ψ(y) ≤ ψ(γ_{x}(s))) + A_{s,t}^{ω}(γ)

and

K_{s,t}^{ω}ψ(x) = ψ(γ_{x}(s))) + A_{s,t}^{ω}(γ_{x})

we have

K_{s,t}^{ω}ψ(y) − K_{s,t}^{ω}ψ(x) ≤ A_{s,t}^{ω}(γ) − A_{s,t}^{ω}(γ_{x}).  (57)

Using (16) one easily gets the following estimate

|A_{s,t}^{ω}(γ) − A_{s,t}^{ω}(γ_{x})| ≤ ((2 + (t − s) max_{τ∈[s,t]} |γ̇_{x}|)∥F ∥_{s,t}^{ω} + |b|) dist(y, x).  (58)

By Lemma 6, max_{τ∈[s,t]} |γ̇_{x}| ≤ C(ω, s, t), which gives

|A_{s,t}^{ω}(γ) − A_{s,t}^{ω}(γ_{x})| ≤ c(ω, s, t)d(y, x),  (59)

where

c(ω, s, t) = (2 + (t − s)C(ω, s, t))∥F ∥_{s,t}^{ω} + |b|.  (60)

Interchanging x and y we obtain

|K_{s,t}^{ω}ψ(y) − K_{s,t}^{ω}ψ(x)| ≤ c(ω, s, t)d(y, x).  (61)

To prove the second statement notice that the estimate (61) does not depend on ψ. Hence, for all s ≤ t − 1

|K_{s,t}^{ω}ψ(y) − K_{s,t}^{ω}ψ(x)| ≤ c(ω, t)d(y, x),  (62)

where c(ω, t) = c(ω, t − 1, t).

Proof of Theorem 1. To prove statement 1 fix arbitrary x_{0} ∈ T^{d}. Obviously there exists a unique function ψ(x, t_{0}) such that ψ(x_{0}, t_{0}) = 0 and ψ̂(t_{0}) = {ψ_{1} = ψ(x, t_{0}) + const}, where ψ̂(t) is the unique solution of the factorized Hamilton-Jacobi equation. Notice, that K_{t,t_{0}}^{ω,b} commutes with an addition of constant. Namely,

K_{t,t_{0}}^{ω,b}(ψ + c) = K_{t,t_{0}}^{ω,b}ψ + c.  (63)

This implies that for any t < t_{0} there exists a unique function ψ(x, t) such that ψ̂(t) = {ψ_{1} = ψ(x, t) + const} and

K_{t,t_{0}}^{ω,b}ψ(x, t) = ψ(x, t_{0}).  (64)

Let us show that ϕ(x, t) = b · x + ψ(x, t) is a solution of the Hamilton-Jacobi equation. Indeed, since ψ̂(t) is the solution of the factorized Hamilton-Jacobi equation we have

K_{t_{1},t_{2}}^{ω,b}ψ(x, t_{1}) = ψ(x, t_{2}) + c(t_{1}, t_{2}),  (65)

where t_{1} < t_{2} < t_{0}. Using (63) - (65) one gets

ψ(x, t_{0}) = K_{t_{1},t_{0}}^{ω,b}ψ(x, t_{1}) = K_{t_{2},t_{0}}^{ω,b}K_{t_{1},t_{2}}^{ω,b}ψ(x, t_{1})
= K_{t_{2},t_{0}}^{ω,b}(ψ(x, t_{2}) + c(t_{1}, t_{2})) = ψ(x, t_{0}) + c(t_{1}, t_{2}),

which implies c(t_{1}, t_{2}) = 0. Uniqueness (up to additive constant) of the solution ψ(x, t) follows easily from uniqueness for factorized Hamilton-Jacobi equation and (63).

We next prove statement 2. Lemma 11 implies that the solution of the Hamilton-Jacobi equation is Lipschitz. Let x be be a point of differentiability of ψ_{b}^{ω}(x, t) and γ_{x,t} be a one-sided minimizer at (x, t). It follows from Lemma 9 that γ_{x,t} is a ψ_{b}^{ω}−minimizer for some s < t. Fix arbitrary s < t. Let x(ϵ), ϵ ∈ (−ϵ_{0}, ϵ_{0}), be a smooth curve in 𝕋^{d} such that x(o) = x, and γ_{ϵ}(τ), ϵ ∈ (−ϵ_{0}, ϵ_{0}), be a smooth family of curves such that γ_{0}(τ) = γ_{x,t}(τ) for all τ ∈ [s, t], γ_{ϵ}(s) = γ_{x,t}(s) and γ_{ϵ}(t) = x(ϵ). Then

ψ_{b}^{ω}(x(ϵ), t) ≤ ψ_{b}^{ω}(γ_{x,t}(s), s) + A_{s,t}^{ω,b}(γ_{ϵ})

and

ψ_{b}^{ω}(x, t) = ψ_{b}^{ω}(γ_{x,t}(s), s) + A_{s,t}^{ω,b}(γ_{x,t})

Hence for positive ϵ,

\frac{ψ_{b}^{ω}(x(ϵ), t) − ψ_{b}^{ω}(x, t)}{ϵ} ≤ \frac{1}{ϵ}(A_{s,t}^{ω,b}(γ_{ϵ}) − A_{s,t}^{ω,b}(γ_{x,t})).  (66)

Taking limit as ϵ ↓ 0 and using differentiability of ψ_{b}^{ω} at x and Lemma 10 we get

∇ψ_{b}^{ω}(x, t) · \frac{dx}{dϵ}(0) ≤ (γ̇_{x,t}(t) − b) · \frac{dx}{dϵ}(0) .  (67)

Analogously, for negative ϵ ↑ 0 we get

∇ψ_{b}^{ω}(x, t) · \frac{dx}{dϵ}(0) ≥ (γ̇_{x,t}(t) − b) · \frac{dx}{dϵ}(0) .  (68)

Since \frac{dx}{dϵ}(0) is an arbitrary vector it follows from (67), (68) that

γ̇_{x,t}(t) = b + ∇ψ_{b}^{ω}(x, t).

Finally, uniqueness of a one-sided minimizer at (x, t) follows from the fact that any minimizer is uniquely determined by its position and velocity at time t.

It remains to prove statement 3. The convergence in

lim_{s→−∞} sup_{η∈C(𝕋^{d})} min_{C∈ℝ} max_{x∈𝕋^{d}}|K_{s,t}^{ω,b}η(x) − ψ_{b}^{ω}(x, t) − C| = 0

follows immediately from Proposition 1. To prove

lim_{s→−∞} sup_{η∈C(𝕋^{d})} sup_{γ_{η,s;x,t}^{ω,b}} |γ̇_{η,s;x,t}^{ω,b}(t) − γ̇_{x,t}^{ω,b}(t)| = 0

assume that convergence above does not hold. Then, for some δ > 0 there exists a sequence s_{n} → −∞, η_{n} ∈ C(𝕋^{d}) and γ_{η_{n},s_{n};x,t}^{ω,b} such that

|γ̇_{η_{n},s_{n};x,t}^{ω,b}(t) − γ̇_{x,t}^{ω,b}(t)| ≥ δ.  (69)

Since by Lemma 6 the sequence γ̇_{η_{n},s_{n};x,t}^{ω,b}(t) is bounded, it follows from (69) that there exists a limiting point γ̇ for γ̇_{η_{n},s_{n};x,t}^{ω,b}(t) different from γ̇_{x,t}^{ω,b}(t). Hence γ̇ is a velocity of another one-sided minimizer at (x, t), which contradicts to the uniqueness.

We finish this section with the proof of Proposition 2. Since the “white force” case is similar to the case of “kicked force” when Assumption 2 holds, we shall consider the “kicked” case only. We start with two lemmas. Denote by

O_{δ} = {F (x) ∈ C^{1}(𝕋^{d}) : ∥F ∥_{C^{1}} ≤ δ}

and by

O_{δ}(G) = {F (x) ∈ C^{1}(𝕋^{d}) : ∥F − G∥_{C^{1}} ≤ δ}.

Lemma 12. Let δ_{n} = \frac{1}{n^{2}} and F_{j} ∈ O_{δ_{n}} for all j ∈ [n_{1}, n_{2}], n_{2} − n_{1} = n. Then, for all x, y ∈ 𝕋^{d}

|A_{n_{1},n_{2}}^{b}(X_{x,n_{1};y,n_{2}}) − \frac{b^{2}n}{2}| ≤ \frac{6d}{n} ,  (70)

where X_{x,n_{1};y,n_{2}} is a minimizing sequence with end points at (x, n_{1}), (y, n_{2}).

Proof. Since diam (𝕋^{d}) ≤ \frac{\sqrt{d}}{2}, we can connect any two points by a sequence X̄_{x,n_{1};y,n_{2}}, such that ρ_{b}(x̄_{j+1}, x̄_{j}) ≤ \frac{\sqrt{d}}{2n} for all j ∈ [n_{1}, n_{2} − 1]. Hence,

A_{n_{1},n_{2}}^{b}(X̄_{x,n_{1};y,n_{2}}) ≤ nδ_{n} + \frac{d}{8n} − \frac{b^{2}n}{2} = \frac{d + 8}{8n} − \frac{b^{2}n}{2}.  (71)

It follows from (71) that for any minimizing sequence X_{x,n_{1};y,n_{2}} the following estimate holds

ρ_{b}(x_{j+1}, x_{j}) ≤ \frac{\sqrt{16 + d} + 2}{2n}, j ∈ [n_{1}.n_{2} − 1].  (72)

To prove (72) notice that |v_{j+1} − v_{j}| ≤ δ_{n}, j ∈ [n_{1}, n_{2} − 1]. Hence, if ρ_{b}(x_{j_{0}+1}, x_{j_{0}}) > \frac{\sqrt{16+d}+2}{2n} for some j_{0} then for all j ∈ [n_{1}, n_{2} − 1]

ρ_{b}(x_{j+1}, x_{j}) > \frac{\sqrt{16 + d} + 2}{2n} − nδ_{n} = \frac{\sqrt{16 + d}}{2n}.

The last estimate implies that

A_{n_{1},n_{2}}^{b}(X_{x,n_{1};y,n_{2}}) > −nδ_{n} + \frac{16 + d}{8n} − \frac{b^{2}n}{2} ≥ A_{n_{1},n_{2}}^{b}(X̄_{x,n_{1};y,n_{2}}).

This contradiction proves (72). It follows that

|A_{n_{1},n_{2}}^{b}(X_{x,n_{1};y,n_{2}}) − \frac{b^{2}n}{2}| ≤ \frac{(\sqrt{16 + d} + 2)^{2}}{8n} + nδ_{n} ≤ \frac{6d}{n} .

Consider again an interval [n_{1}, n_{2}], n_{2} − n_{1} = n. Let z_{j} ∈ 𝕋^{d}, j ∈ [n_{1}, n_{2}] be a sequence of points such that z_{j+1} = z_{j} + b̄ (mod 1). Denote by G_{j}, j ∈ [n_{1}, n_{2} − 1], a sequence of potentials which have a unique global maximum at z_{j}. We shall also assume that G_{j} satisfy the condition (29).

Lemma 13. Let β_{n} = \frac{1}{n}, δ_{n} = \frac{1}{n^{2}} and F_{j} ∈ O_{δ_{n}}(G_{j}) for all j ∈ (n_{1}, n_{2}). Then for any α > 0 there exists N(α) such that for all n ≥ N(α), all b ∈ O_{δ_{n}}(b̄) = {b : |b − b̄| ≤ β_{n}} and for all minimizing sequences X_{x,n_{1};y,n_{2}} for the actions A_{n_{1},n_{2}}^{b} the following estimate holds

dist(x_{j}, z_{j}) ≤ α, n_{1} + [n/3] < j < n_{2} − [n/3],

where [·] stands for an integer part.

Proof. Assume without loss of generality that max_{x∈𝕋^{d}} G_{j}(x) = 0, j ∈ (n_{1}, n_{2}). For arbitrary x, y ∈ 𝕋^{d} let X̄ be the following sequence connecting (x, n_{1}) with (y, n_{2}) :

X̄(n_{1}) = x, X̄(j) = z_{j}, n_{1} < j < n_{2}, X̄(n_{2}) = y.

Then

A_{n_{1},n_{2}}^{b}(X̄) ≤ 2\frac{(diam (𝕋^{d}))^{2}}{2} + \frac{(n − 2)β_{n}^{2}}{2} + (n − 1)δ_{n} − F_{n_{1}}(x_{n_{1}}) − \frac{b^{2}n}{2}
≤ \frac{d}{4} + \frac{3}{2n} − F_{n_{1}}(x_{n_{1}}) − \frac{b^{2}n}{2}.  (73)

Consider now a minimizing sequence X_{x,n_{1};y,n_{2}}. Fix ρ > 0 such that ρ^{2} < a(α). (see condition (29)). Obviously, F_{j}(x) ≥ a(ρ)−δ_{n} for all x such that dist(x, z_{j}) ≥ ρ. Denote by R the number of j′s, j ∈ (n_{1}, n_{2}) such that dist(x_{j}, z_{j}) ≥ ρ. Then the following estimate holds

A_{n_{1},n_{2}}^{b}(X_{x,n_{1};y,n_{2}}) ≥ R(a(ρ) − δ_{n}) − (n − R − 1)δ_{n} − F_{n_{1}}(x_{n_{1}}) − \frac{b^{2}n}{2}.  (74)

Let n be so large that

\frac{\frac{5}{2n} + \frac{d}{4}}{a(ρ)} < [\frac{n}{3}] .  (75)

It follows from (73) - (75) that

R ≤ \frac{(n − 1)δ_{n} + \frac{3}{2n} + \frac{d}{4}}{a(ρ)} < [\frac{n}{3}] .

This implies that there exist m_{1}, m_{2}, n_{1}+1 ≤ m_{1} ≤ n_{1}+[n/3], n_{2}−[n/3] ≤ m_{2} ≤ n_{2}−1 such that

dist(x_{m_{1}}, z_{m_{1}}), dist(x_{m_{2}}, z_{m_{2}}) ≤ ρ .

Finally, we show that dist(x_{j}, z_{j}) ≤ α for all j ∈ (m_{1}, m_{2}) provided n is so large that

a(α) > (ρ + \frac{1}{n})^{2} + \frac{5}{2n} .  (76)

Indeed, if dist(x_{j}, z_{j}) > α for some j ∈ (m_{1}, m_{2}) then

A_{m_{1},m_{2}}^{b}(X_{x,n_{1};y,n_{2}}) ≥ (a(α) − δ_{n}) − (m − 2)δ_{n} − F_{m_{1}}(x_{m_{1}}) − \frac{b^{2}m}{2},  (77)

where m = m_{2} − m_{1}. At the same time for the sequence

X̃(m_{1}) = x_{m_{1}}, X̃(j) = z_{j}, m_{1} < j < m_{2}, X̃ = x_{m_{2}}

one has

A_{m_{1},m_{2}}^{b}(X̃) ≤ 2\frac{(ρ + β_{n})^{2}}{2} + \frac{(m − 2)β_{n}^{2}}{2} + (m − 1)δ_{n} − F_{m_{1}}(x_{m_{1}}) − \frac{b^{2}m}{2}.  (78)

Using (76) - (78) we obtain

A_{m_{1},m_{2}}^{b}(X̃) < A_{m_{1},m_{2}}^{b}(X_{x,n_{1};y,n_{2}}),

which contradicts to the fact that X_{x,n_{1};y,n_{2}} is a minimizer. Hence, dist(x_{j}, z_{j}) ≤ α for all j ∈ (m_{1}, m_{2}).

Proof of Proposition 2. We shall first deal with the case when Assumption 2 holds. Fix arbitrary n ∈ ℕ. For an interval [n_{1}, n_{2}], n_{2} − n_{1} = n consider the following event:

F_{j} ∈ O_{δ_{n}} for all j ∈ [n_{1}, n_{2}],  (79)

where δ_{n} = \frac{1}{n^{2}} . Since zero potential F_{0}(x) ≡ 0 belongs to the supp (χ), an event (79) has positive probability. Hence, with probability 1 there exists an infinite sequence of intervals [n_{1}^{(k)}, n_{2}^{(k)}] satisfying (79) and such that n_{1}^{(k)} → −∞ as k → ∞. It follows that with probability 1 such sequence of intervals [n_{1}^{(k)}, n_{2}^{(k)}] exists for all n. Lemma 12 immediately implies that for a given n each interval [n_{1}^{(k)}, n_{2}^{(k)}] form an ϵ−narrow place with ϵ = \frac{12d}{n} and s_{1} = n_{1}^{(k)}, s_{2} = n_{2}^{(k)}, M_{1} = 𝕋^{d}, M_{2} = 𝕋^{d}. This proves statement 2 in the case of Assumption 2.

We next consider the case of Assumption 1. Fix arbitrary x̄ ∈ 𝕋^{d}. For all x ∈ 𝕋^{d} denote by G = G_{x} a potential with unique global maximum in x which belongs to supp(χ) and satisfies condition (29). Let B̄ be a countable dense set of vectors b̄ in ℝ^{d}. For an arbitrary n ∈ ℕ, b̄ ∈ B̄ and interval [n_{1}, n_{2}], n_{2} − n_{1} = n put δ_{n} = \frac{1}{n^{2}} and consider the following event:

F_{j} ∈ O_{δ_{n}}(G_{z_{j}}) for all j ∈ [n_{1}, n_{2}],  (80)

where z_{j} ∈ 𝕋^{d}, j ∈ [n_{1}, n_{2}] is a sequence of points such that z_{j+1} = z_{j} + b̄ (mod 1) and z_{n_{1}+[n/2]} = x̄. As above, this event has positive probability. Hence, with probability 1 for all n ∈ ℕ, b̄ ∈ B̄ there exists an infinite sequence of intervals [n_{1}^{(k)}, n_{2}^{(k)}] satisfying (80) and such that n_{1}^{(k)} → −∞ as k → ∞. Let interval [n_{1}, n_{2}], n_{2}−n_{1} = n satisfies (80) and n ≥ N(α), where N(α) is defined in Lemma 13. Take s_{1} = n_{1}+[n/2], s_{2} = s_{1}+1, M_{1} = {x : dist(x, z_{s_{1}}) ≤ α}, M_{2} = {x : dist(x, z_{s_{2}}) ≤ α}. It follows from Lemma 13 that for all b ∈ O_{β_{n}}(b̄) all minimizers for actions A_{n_{1},n_{2}}^{b} pass through M_{1}, M_{2} at times s_{1}, s_{2} respectively. Notice that z_{s_{1}} = x̄. Denote by c(α) = max_{dist(x,x̄)≤α}(Gx̄(x̄) − Gx̄(x)). It follows from Lemma 13 that {[n_{1}, n_{2}], s_{1}, M_{1}, s_{2}, M_{2}} form an ϵ−narrow place with

ϵ = (2α + β_{n})^{2} + 2c(α).  (81)

Indeed, for any x ∈ M_{1}, y ∈ M_{2} we have A_{s_{1},s_{2}}^{b}\{x, y\} = \frac{(ρ_{b}(x,y))^{2}}{2} − Gx̄(x) − \frac{b^{2}}{2} . Hence,

|A_{s_{1},s_{2}}^{b}\{x, y\} + Gx̄(x̄) + \frac{b^{2}}{2} | ≤ \frac{(2α + β_{n})^{2}}{2} + c(α),

which gives (81). Since c(α) → 0 as α → 0, the estimate (81) implies the statement of Proposition 2. To construct an ϵ−narrow place for arbitrary ϵ > 0, b ∈ ℝ^{n} we first choose α so small that 2c(α) + (2α)^{2} ≤ ϵ/2, then take n ≥ N(α) so large that 4αβ_{n} + β_{n}^{2} ≤ ϵ/2 and finally pick one of the intervals [n_{1}, n_{2}], n_{2} − n_{1} = n satisfying (80) for some b̄ in the the β_{n}−neighbourhood of b.

In the case of Assumption 3 the proof is similar.

6. Stationary distribution for the random Burgers equation

In this section we prove Lemmas 1, 2 and Theorem 3. We consider only the “white force” case since in the “kicked case” the proofs are similar.

Proof of Lemma 1. Denote by B_{b}^{p} the σ−algebra on U_{b} generated by (L^{p}(𝕋^{d}, dx), B^{p}), 1 ≤ p < ∞ :

B_{b}^{p} = {A ∈ U_{b} : A = U_{b} ∩ B, B ∈ B^{p}},  (82)

where B^{p} are Borel σ−algebras on L^{p}(𝕋^{d}, dx). We shall show that

B_{b}^{p} = B_{b}^{1} for all p > 1.  (83)

Fix arbitrary p > 1. Denote by B̄ the σ−algebra on L^{p}(𝕋^{d}, dx) which is generated by L^{1}(𝕋^{d}, dx) :

B̄ = {A ∈ L^{p}(𝕋^{d}, dx) : A = L^{p}(𝕋^{d}, dx) ∩ B, B ∈ B^{1}}.

To prove (83) it is enough to show that B̄ = B^{p} . Notice, that B̄ is the minimal σ−algebra which contains all sets Ū = L^{p}(𝕋^{d}, dx) ∩ U, where U is an open subset of L^{1}(𝕋^{d}, dx). It follows immediately that Ū is open in L^{p}(𝕋^{d}, dx). Hence

B̄ ⊆ B^{p} .  (84)

On the other hand, since L^{p}(𝕋^{d}, dx) is a separable Banach space any open set Ũ ∈ L^{p}(𝕋^{d}, dx) is a countable union of closed balls: Ũ = ∪B_{i} . We next show that any closed ball B_{i} in L^{p}(𝕋^{d}, dx) is a closed set in L^{1}(𝕋^{d}, dx). Indeed, assume that a sequence f_{n} ∈ B_{i} converges to g in L^{1}(𝕋^{d}, dx) : ∥f_{n} − g∥_{L^{1}} → 0 as n → ∞. This implies that for any h ∈ L^{∞}(𝕋^{d}, dx)

∫_{𝕋^{d}}(f_{n} · h)dx → ∫_{𝕋^{d}}(g · h)dx as n → ∞.  (85)

Since a closed ball in L^{p}(𝕋^{d}, dx), p > 1 is compact in weak-topology, there exists a subsequence f_{n_{k}} which converges weakly to f ∈ B_{i}. It follows that for all h ∈ L^{∞}(𝕋^{d}, dx)

∫_{𝕋^{d}}(f_{n_{k}} · h)dx → ∫_{𝕋^{d}}(f · h)dx as k → ∞.  (86)

Using (85), (86) we get

∫_{𝕋^{d}}(g · h)dx = ∫_{𝕋^{d}}(f · h)dx ,

which implies g = f (a.s.). Hence g ∈ B_{i} which proves that B_{i} is closed in L^{1}(𝕋^{d}, dx). It follows immediately that all B_{i} and, hence, Ũ belong to B̄. Thus,

B^{p} ⊆ B̄ .  (87)

Using (84), (87) we finally get B̄ = B^{p}.

Proof of Lemma 2. For a sequence −T_{n} → −∞ as n → ∞ denote by u_{b,−T_{n}}^{ω}(x) = u_{b,−T_{n}}^{ω}(x, 0) the solution of the Cauchy problem for Burgers equation on [−T_{n}, 0] with the initial value u_{b,−T_{n}}^{ω}(x, −T_{n}) ≡ 0. We shall regard u_{b,−T_{n}}^{ω}(x) as a “random variable” taking values in U_{b} ⊂ L^{1}(𝕋^{d}). Clearly, for every finite T_{n} u_{b,−T_{n}}^{ω} corresponds to a measurable mapping from (Ω, F) into (U_{b}, B_{b}^{1}), where B_{b}^{1} is defined in (82). It follows from Theorem 1 and Lemma 6 that for almost all ω

∥u_{b,−T_{n}}^{ω}(x) − u_{b}^{ω}(x)∥_{L^{1}} → 0 as n → ∞.  (88)

This relation means that “random variables” u_{b,−T_{n}}^{ω} converge almost surely to u_{b}^{ω}. Using (88) and completeness of a measure space (Ω, F, P) we conclude that u_{b}^{ω} defines a measurable mapping from (Ω, F) into (U_{b}, B_{b}^{1}). To finish the proof it is enough to notice that B_{b}^{1} = B_{b} (see Lemma 1).

Proof of Theorem 3. We shall start with the proof of the first statement. It follows from the construction of δ_{b}^{ω}(du) that

B_{∗}^{ω}(τ) δ_{b}^{ω}(du) = δ_{b}^{θ^{τ} ω}(du), τ > 0,

which immediately implies invariance of the measure ν_{b}(du, dω) = δ_{b}^{ω}(du)P (dω). To prove uniqueness suppose that ν(du, dω) = ν^{ω}(du)P (dω) ∈ P(U_{b} × Ω) is an arbitrary invariant measure for B̂(τ), τ > 0 with the marginal distribution P (dω) on Ω. Then for all τ > 0 and P −almost all ω ∈ Ω we have

B_{∗}^{θ^{−τ} ω}(τ) ν^{θ^{−τ} ω}(du) = ν^{ω}(du).

It follows that for P −almost all ω ∈ Ω there exists a subset U^{ω} ⊆ U_{b} of full ν^{ω}−measure such that for all n > 0 and all u ∈ U^{ω}

B^{θ^{−n}ω}(n)u_{n}(x) = u(x)

for some u_{n}(x) ∈ U_{b}. Using statement 3 of Theorem 1 we conclude that u(x) = u_{b}^{ω}(x, 0) for Lebesgue-almost all x ∈ 𝕋^{d}. Hence, ν^{ω}(du) = δ_{b}^{ω}(du) and ν = ν_{b}.

We next prove the second statement. Since δ_{b}^{ω}(du) is measurable with respect to σ - algebra F_{−∞}^{0} and measure ν_{b} is invariant for for the skew-product dynamics (7) it is easy to show that the measure

μ_{b}(du) = ∫_{Ω} δ_{b}^{ω}(du)P (dω)

gives a stationary distribution for the Markov process (8). Let μ ∈ P(U_{b}) be another stationary distribution. Consider arbitrary bounded continuous function C(u) on L^{1}(𝕋^{d}, dx). Then,

∫_{U_{b}} C(u)μ(du) = ∫_{U_{b}} [∫_{Ω} C(B^{θ^{−n}ω}(n)ū)P (dω)] μ(dū).

Theorem 1 and Lemma 6 imply that for all ū and P −almost all ω

∥B^{θ^{−n}ω}(n)ū − u_{b}^{ω}(x, 0)∥_{L^{1}} → 0 as n → ∞.

Since C(u) is continuous in L^{1}(𝕋^{d}) and bounded we get

C(B^{θ^{−n}ω}(n)ū) → C(u_{b}^{ω}(x, 0)) as n → ∞

and

∫_{Ω} C(B^{θ^{−n}ω}(n)ū)P (dω) → ∫_{Ω} C(u_{b}^{ω}(x, 0))P (dω).

Hence, in a limit n → ∞ we have

∫_{U_{b}} [∫_{Ω} C(B^{θ^{−n}ω}(n)ū)P (dω)] μ(dū) → ∫_{U_{b}} [∫_{Ω} C(u_{b}^{ω}(x, 0))P (dω)] μ(dū) =

∫_{Ω} C(u_{b}^{ω}(x, 0))P (dω) = ∫_{U_{b}} C(u)μ_{b}(du).

It follows that

∫_{U_{b}} C(u)μ(du) = ∫_{U_{b}} C(u)μ_{b}(du)

and μ = μ_{b}.

Remark 6.1.

1. Notice that in the proof of the first statement of Theorem 3 we did not use condition 3 of Definition 1. Hence the invariant measure ν_{b} is unique in a class of all invariant measure for the skew-product dynamics with the marginal distribution on Ω equal to P (dω).

2. We gave an independent proof of the second statement of Theorem 3 although it can be obtained from the first statement. Indeed, it follows from [1, 13] that there exists a one-to-one correspondence between stationary distributions μ_{b}(du) and invariant measures ν_{b}(du, dω) which satisfy condition 3 of Definition 1.

7. Uniqueness of the Global Minimizer

In this section we shall prove Theorems 4 and 5. The idea of the proof is similar in both cases, but in the “white force” case the proof has some technical difficulties. We start with the extension of the results of section 5 to the forward minimizers. More precisely, in section 5 we were concerned with one-sided minimizers ending at some point x_{0} at time t_{0}. In this section we shall call such curves backward minimizers. It was shown in section 5 that backward minimizers are related to a unique solution of the Hamilton-Jacobi equation in a semi-infinite interval of time (−∞, t_{0}]. The following definition is similar to Definition 3.

Definition 5. A curve γ_{x_{0},t_{0}}^{+} : [t_{0}, ∞) → 𝕋^{d} is called a forward minimizer at a point (x_{0}, t_{0}) for the action A^{ω,b} if it is a minimizer for all t_{0} ≤ s < t < ∞ and γ_{x_{0},t_{0}}^{+}(t_{0}) = x_{0}.

The same definition applies to forward minimizing sequences for the action A^{ω,b}. We shall also define the forward Lax operators

K_{s,t}^{+,ω,b}ψ(x) = inf_{γ∈AC(x,s;t)} (ψ(γ(t)) + A_{s,t}^{ω,b}(γ)) ,

where AC(x, s; t) is the set of all absolutely continuous curves γ : [s, t] → 𝕋^{d} such that γ(s) = x. In the “kicked” case the definition of LAX operators is similar:

K_{m,n}^{+,ω,b}ψ(x) = inf_{X∈S(x,m;n)} (ψ(x_{n}) + A_{m,n}^{ω,b}(X)) ,

where S(x, m; n) is the set of sequences X = {x_{i}, m ≤ i ≤ n, x_{i} ∈ 𝕋^{d}} such that x_{m} = x. As it explained below Theorems 1, 2 remain valid for forward minimizers and minimizing sequences if we replace K_{s,t}^{ω,b}, K_{m,n}^{ω,b} by K_{s,t}^{+,ω,b}, K_{m,n}^{+,ω,b}. In particular, this implies that for P −almost all ω and all b there exist a unique Lipschitz function ψ_{b}^{+,ω}(x, t) such that for all s < t

ψ_{b}^{+,ω}(x, s) = K_{s,t}^{+,ω,b}ψ_{b}^{+,ω}(x, t).

In the “kicked” case ψ_{b}^{+,ω}(x, t) is replaced by ψ_{b}^{+,ω}(x, n±), ψ_{b}^{+,ω}(x, n−) = ψ_{b}^{+,ω}(x, n+)− F_{n}^{ω}(x). Then for all m < n we have the following equation:

ψ_{b}^{+,ω}(x, m+) = K_{m,n}^{+,ω,b}ψ_{b}^{+,ω}(x, n+).

As in the “backward” case, functions ψ_{b}^{+,ω}(x, t), ψ_{b}^{+,ω}(x, n±) are related to the velocities of the forward minimizers. Namely, if x is a point of differentiability of ψ_{b}^{+,ω}(x, t) (ψ_{b}^{+,ω}(x, n+)) then then there exists a unique forward minimizer (minimizing sequence) at (x, t) ((x, n)) and its velocity at time t (n+) is given by b−∇ψ_{b}^{+,ω}(x, t) (b− ∇ψ_{b}^{+,ω}(x, n+)). All the statements above follow from the results of section 5. The best way to see this is to apply an automorphism R of the measure space (Ω, F, P) corresponding to the time-reversing transformation t → −t :

F^{Rω}(x, t) = F^{ω}(x, −t).

Obviously, R is a measure-preserving automorphism in both “white force” and “kicked” cases. It is easy to see that

ψ_{b}^{+,ω}(x, t) = ψ_{−b}^{Rω}(x, −t), ψ_{b}^{+,ω}(x, n±) = ψ_{−b}^{Rω}(x, −n∓)

and

γ_{x_{0},t_{0}}^{+,ω,b}(τ) = γ_{x_{0},−t_{0}}^{Rω,−b}(−τ), t_{0} ≤ τ < ∞, X_{x_{0},n_{0}}^{+,ω,b}(n) = X_{x_{0},−n_{0}}^{Rω,−b}(−n), n_{0} ≤ n < ∞.

It follows immediately that all the statements of section 5 remain valid with obvious changes: ω → Rω, b → −b, t → −t. Notice that ϕ_{b}^{+,ω}(y, t) = −b · y + ψ_{b}^{+,ω}(y, t) satisfy the “backward” Hamilton-Jacobi equation:

−∂_{t}ϕ(y, t) + \frac{1}{2}(∇ϕ(y, t))^{2} + F^{ω}(y, t) = 0.

We shall also use the existence of ϵ−narrow places in the past and in the future. More precisely, Proposition 2 implies that there exists a subset Ω_{1} ⊂ Ω of full P −measure such that for any ω ∈ Ω_{1}, any ϵ > 0, b ∈ ℝ^{d} and any t < t̄ there exist ϵ−narrow places [t_{1}, t_{2}] ⊂ (−∞, t], [t̄_{1}, t̄_{2}] ⊂ [t̄, ∞).

To make notations more symmetric we shall denote ψ_{b}^{ω}(x, t), ψ_{b}^{ω}(x, n±) by ψ_{b}^{−,ω}(x, t), ψ_{b}^{−,ω}(x, n±). We also denote backward minimizers by γ_{x_{0},t_{0}}^{−,ω,b}(τ), X_{x_{0},n_{0}}^{−,ω,b}(n). As before we denote by γ_{(x_{1},s;x_{2},t)}^{ω,b}(τ), τ ∈ [s, t] any minimizer connecting points (x_{1}, s) and (x_{2}, t), s < t. In the “kicked” case we denote minimizing sequence connecting (x_{1}, m) and (x_{2}, n), m < n by X_{(x_{1},m;x_{2},n)}^{ω,b}. Below we also use the following notations

A_{s,t}^{ω,b}(x_{1}.x_{2}) = A_{s,t}^{ω,b}(γ_{(x_{1},s;x_{2},t)}^{ω,b}), A_{m,n}^{ω,b}(x_{1}.x_{2}) = A_{m,n}^{ω,b}(X_{(x_{1},m;x_{2},n)}^{ω,b}).

Obviously,

A_{s,t}^{ω,b}(x_{1}.x_{2}) = min_{γ∈AC(x_{1},s;x_{2},t)} A_{s,t}^{ω,b}(γ), A_{m,n}^{ω,b}(x_{1}.x_{2}) = min_{X∈S(x_{1},m;x_{2},n)} A_{m,n}^{ω,b}(X).

Since b in this section is fixed we shall omit it in the notations whenever possible. The following proposition provides a useful condition for global minimizers and global minimizing sequences.

Proposition 3. Suppose ω belongs to Ω_{1}. Fix arbitrary s ≤ t, s, t ∈ ℝ and m ≤ n, m, n ∈ ℤ. Then the following statements hold.

1. A curve γ(τ), τ ∈ ℝ is a global minimizer for an action A^{ω} if and only if γ(τ), τ ∈ (−∞, s] and γ(τ), τ ∈ [t, ∞) are backward and forward minimizers, γ(τ), τ ∈ [s, t] is a minimizer connecting points (γ(s), s), (γ(t), t) and

ψ^{−,ω}(γ(s), s) + ψ^{+,ω}(γ(t), t) + A_{s,t}^{ω}(γ(s), γ(t)) = min_{x_{1},x_{2}∈𝕋^{d}×𝕋^{d}}[ψ^{−,ω}(x_{1}, s) +

ψ^{+,ω}(x_{2}, t) + A_{s,t}^{ω}(x_{1}, x_{2})].  (89)

2. A sequence X = {x_{i}, i ∈ ℤ} is a global minimizing sequence if and only if {x_{i}, i ≤ m} and {x_{i}, i ≥ n} are backward and forward minimizing sequences, {x_{i}, i ∈ [m, n]} is a minimizing sequence connecting points (x_{m}, m), (x_{n}, n) and

ψ^{−,ω}(x_{m}, m−) + ψ^{+,ω}(x_{n}, n−) + A_{m,n}^{ω}(x_{m}, x_{n}) = min_{x_{1},x_{2}∈𝕋^{d}×𝕋^{d}}[ψ^{−,ω}(x_{1}, m−) +

ψ^{+,ω}(x_{2}, n−) + A_{m,n}^{ω}(x_{1}, x_{2})]. (90)

Remark 7.1. Intuitively one can think of ψ^{−,ω}(x, s), ψ^{+,ω}(x, t) as the action of the backward and forward minimizers at (x, s) and (x, t) for infinitely long intervals of time (−∞, s] and [t, ∞). The same interpretation is also helpful in the “kicked” case. Notice, that as random variables ψ^{−,ω}(x, s), ψ^{+,ω}(x, t) and ψ^{−,ω}(x, m±), ψ^{+,ω}(x, n±) are statistically independent.

Proof of Proposition 3. We shall give a proof for the case of “white force”. In the “kicked” case the proof is similar. Consider a global minimizer γ(τ), τ ∈ ℝ. Obviously, it consists of three minimizing pieces: γ(τ), τ ∈ (−∞, s] and γ(τ), τ ∈ [t, ∞) are backward and forward minimizers, γ(τ), τ ∈ [s, t] is a minimizer connecting points (γ(s), s), (γ(t), t). Suppose that γ does not satisfy (89). Then for some x, y ∈ 𝕋^{d} and ϵ > 0 we have

ψ^{−,ω}(x, s) + ψ^{+,ω}(y, t) + A_{s,t}^{ω}(x, y) + ϵ < ψ^{−,ω}(γ(s), s) + ψ^{+,ω}(γ(t), t) +
A_{s,t}^{ω}(γ(s), γ(t)) .  (91)

Define σ(τ), τ ∈ ℝ by joining the following minimizers:

σ(τ) = \begin{cases}γ_{x,s}^{−,ω}(τ)  for τ ∈ (−∞, s] \\ γ_{(x,s;y,t)}^{ω}(τ) for τ ∈ [s, t] \\ γ_{y,t}^{+,ω}(τ)  for τ ∈ [t, ∞) . \end{cases}

Let [t_{1}^{−}, t_{2}^{−}] ⊂ (−∞, s] and [t_{1}^{+}, t_{2}^{+}] ⊂ [t, ∞) be \frac{ϵ}{10}−narrow places and [s_{1}^{−}, s_{2}^{−}] ⊂ [t_{1}^{−}, t_{2}^{−}], [s_{1}^{+}, s_{2}^{+}] ⊂ [t_{1}^{+}, t_{2}^{+}] be corresponding subintervals from Definition 4. Using Lemma 9 and (42) we obtain

|ψ^{−,ω}(σ(s_{1}^{−}), s_{1}^{−}) − ψ^{−,ω}(γ(s_{1}^{−}), s_{1}^{−})| ≤ \frac{ϵ}{10} ,

and

|ψ^{+,ω}σ(s_{2}^{+}), s_{2}^{+}) − ψ^{+,ω}(γ(s_{2}^{+}), s_{2}^{+}) ≤ \frac{ϵ}{10} .

Hence,

|ψ^{−,ω}(σ(s_{2}^{−}), s_{2}^{−}) − ψ^{−,ω}(γ(s_{2}^{−}), s_{2}^{−})| ≤ \frac{3ϵ}{10} ,

and

|ψ^{+,ω}σ(s_{1}^{+}), s_{1}^{+}) − ψ^{+,ω}(γ(s_{1}^{+}), s_{1}^{+})| ≤ \frac{3ϵ}{10} .

Moreover by Lemma 9 the following relations hold

ψ^{−,ω}(γ(s), s) = ψ^{−,ω}(γ(s_{2}^{−}), s_{2}^{−}) + A_{s_{2}^{−},s}^{ω}(γ),
ψ^{+,ω}(γ(t), t) = ψ^{+,ω}(γ(s_{1}^{+}), s_{1}^{+}) + A_{t,s_{1}^{+}}^{ω}(γ),
ψ^{−,ω}(σ(s), s) = ψ^{−,ω}(σ(s_{2}^{−}), s_{2}^{−}) + A_{s_{2}^{−},s}^{ω}(σ),
ψ^{+,ω}(σ(t), t) = ψ^{−,ω}(σ(s_{1}^{+}), s_{1}^{+}) + A_{t,s_{1}^{+}}^{ω}(σ) .

Using inequality (91) we conclude

A_{s_{2}^{−},s_{1}^{+}}^{ω}(γ) = A_{s_{2}^{−},s}^{ω}(γ) + +A_{s,t}^{ω}(γ(s), γ(t)) + A_{t,s_{1}^{+}}^{ω}(γ)
= ψ^{−,ω}(γ(s), s) − ψ^{−,ω}(γ(s_{2}^{−}), s_{2}^{−}) + A_{s,t}^{ω}(γ(s), γ(t)) +

ψ^{+,ω}(γ(t), t) − ψ^{+,ω}(γ(s_{1}^{+}), s_{1}^{+})

> ψ^{−,ω}(σ(s), s) − ψ^{−,ω}(γ(s_{2}^{−}), s_{2}^{−}) + A_{s,t}^{ω}(x, y) +

ψ^{+,ω}(σ(t), t) − ψ^{+,ω}(γ(s_{1}^{+}), s_{1}^{+}) + ϵ

≥ ψ^{−,ω}(σ(s), s) − ψ^{−,ω}(σ(s_{2}^{−}), s_{2}^{−}) + A_{s,t}^{ω}(σ(s), σ(t)) +

ψ^{+,ω}(σ(t), t) − ψ^{+,ω}(σ(s_{1}^{+}), s_{1}^{+}) + ϵ − \frac{3ϵ}{5}

= A_{s_{2}^{−},s_{1}^{+}}^{ω}(σ) + \frac{2ϵ}{5} .

Finally, taking a curve

σ̄(τ) = \begin{cases}γ_{(γ(s_{1}^{−}),s_{1}^{−};σ(s_{2}^{−}),s_{2}^{−})}^{ω}(τ)  for τ ∈ [s_{1}^{−}, s_{2}^{−}]] \\ σ(τ)  for τ ∈ [s_{2}^{−}, s_{1}^{+}] \\ γ_{(σ(s_{1}^{+}),s_{1}^{+};γ(s_{2}^{+}),s_{2}^{+})}^{ω}(τ)  for τ ∈ [s_{1}^{+}, s_{2}^{+}] \end{cases}

we have

A_{s_{1}^{−},s_{2}^{+}}^{ω}(σ̄) ≤ A_{s_{2}^{−},s_{1}^{+}}^{ω}(σ) + \frac{2ϵ}{10} < A_{s_{2}^{−},s_{1}^{+}}^{ω}(γ) − \frac{2ϵ}{5} + \frac{2ϵ}{10}

≤ A_{s_{1}^{−},s_{2}^{+}}^{ω}(γ) − \frac{2ϵ}{5} + \frac{4ϵ}{10} = A_{s_{1}^{−},s_{2}^{+}}^{ω}(γ) ,

which is a contradiction since γ is a global minimizer.

We next prove that (89) provides a sufficient condition for a curve to be a global minimizer. Suppose γ is not a global minimizer although it consists of three minimizing pieces and satisfy (89). Hence, there exists a curve σ̃(τ), τ ∈ [s̃, t̃] such that s̃ < s, t̃> t, γ(s̃) = σ̃(s̃), γ(t̃) = σ̃(t̃) and A_{s̃,t̃}^{ω}(σ̃) < A_{s̃,t̃}^{ω}(γ). Denote x = σ̃(s), y = σ̃(t). Since

ψ^{−,ω}(x, s) ≤ A_{s̃,s}^{ω}(σ̃) + ψ^{−,ω}(γ(s̃), s̃), ψ^{+,ω}(y, t) ≤ A_{t,t̃}^{ω}(σ̃) + ψ^{+,ω}(γ(t̃), t̃)

we have

ψ^{−,ω}(x, s) + ψ^{+,ω}(y, t) + A_{s,t}^{ω}(x, y) ≤ A_{s̃,s}^{ω}(σ̃) + ψ^{−,ω}(γ(s̃), s̃) + A_{t,t̃}^{ω}(σ̃)
+ψ^{+,ω}(γ(t̃), t̃) + A_{s,t}^{ω}(σ̃) = ψ^{−,ω}(γ(s̃), s̃) + ψ^{+,ω}(γ(t̃), t̃) + A_{s̃,t̃}^{ω}(σ̃)
< ψ^{−,ω}(γ(s̃), s̃) + ψ^{+,ω}(γ(t̃), t̃) + A_{s̃,t̃}^{ω}(γ)
= ψ^{−,ω}(γ(s), s) + ψ^{+,ω}(γ(t), t) + A_{s,t}^{ω}(γ(s), γ(t)) ,

which contradicts to (89).

Remark 7.2. Notice, that Proposition 3 remains valid if s = t or m = n. In this case we have

ψ^{−,ω}(γ(t), t) + ψ^{+,ω}(γ(t), t) = min_{x∈𝕋^{d}}[ψ^{−,ω}(x, t) + ψ^{+,ω}(x, t)]  (92)

instead of (89) and

ψ^{−,ω}(x_{n}, n−) + ψ^{+,ω}(x_{n}, n+) − F_{n}^{ω}(x_{n}) = min_{x∈𝕋^{d}}[ψ^{−,ω}(x, n−) +

ψ^{+,ω}(x, n+) − F_{n}^{ω}(x)]  (93)

instead of (90). Below we use Proposition 3 in the “kicked” case in the form (93).

Next we prove the following general result.

Lemma 14. Let V (x, c), x ∈ 𝕋^{d}, c ∈ ℝ^{k} be a real differentiable function and ψ(x) a continuous function on 𝕋^{d}. Define

G(c) = min_{x∈𝕋^{d}}[ψ(x) + V (x, c)] .

Then the following statements hold.

1. G(c) is a local Lipschitz function.

2. If G(c) is differentiable at c_{0} and x_{0} is a point of minimum corresponding to c_{0} then

\frac{dG}{dc}(c_{0}) = \frac{∂V}{∂c} (x_{0}, c_{0}) .  (94)

Proof. Let x_{1}, x_{2} be points of minimum corresponding to c_{1}, c_{2} respectively. Then

G(c_{1}) = ψ(x_{1}) + V (x_{1}, c_{1})

and

G(c_{2}) ≤ ψ(x_{1}) + V (x_{1}, c_{2})

Subtracting the first equation from the second one we get

G(c_{2}) − G(c_{1}) ≤ V (x_{1}, c_{2}) − V (x_{1}, c_{1}) .

Similarly,

G(c_{1}) − G(c_{2}) ≤ V (x_{2}, c_{1}) − V (x_{2}, c_{2}) .

For any C > 0 define

D(C) = max_{x∈𝕋^{d}} max_{c:|c|≤C} ∥\frac{∂V}{∂c} (x, c)∥ .

Then for all c_{1}, c_{2} such that |c_{1}|, |c_{2}| ≤ C

|G(c_{1}) − G(c_{2})| ≤ D(C)|c_{1} − c_{2}| .

Suppose now that G is differentiable at c_{0} and x_{0} is a point of minimum. As above, for all h ∈ ℝ^{k} and all ϵ ∈ ℝ

G(c_{0} + ϵh) − G(c_{0}) ≤ V (x_{0}, c_{0} + ϵh) − V (x_{0}, c_{0}) .

Dividing by ϵ > 0 and taking limit as ϵ tends to zero we obtain:

\frac{dG}{dc}(c_{0}) · h ≥ \frac{∂V}{∂c} (x_{0}, c_{0}) · h .

In the same way taking limit for negative ϵ we get:

\frac{dG}{dc}(c_{0}) · h ≤ \frac{∂V}{∂c} (x_{0}, c_{0}) · h ,

which immediately implies (94).

Corollary 5. Suppose that for every fixed c ∈ ℝ^{k} the mapping

\frac{∂V}{∂c} (·, c) : 𝕋^{d} → ℝ^{k}

is one-to-one. Then for Lebesgue-almost all c there is a unique point of minimum x(c).

Proof. It follows from Lemma 14 that G(c) = min_{x∈𝕋^{d}}[ψ(x) + V (x, c)] is a Lipschitz function. Hence, by the Rademacher Theorem it is differentiable almost everywhere. It is easy to see that for any point of differentiability there exists a unique point of minimum. Indeed, if c is a point of differentiability and x_{1}(c), x_{2}(c) are two points of minimum we have

\frac{dG}{dc}(c) = \frac{∂V}{∂c} (x_{1}(c), c) = \frac{∂V}{∂c} (x_{2}(c), c) ,

which implies x_{1}(c) = x_{2}(c).

Proof of the Theorem 5. Denote c = (c_{i}, 1 ≤ i ≤ M) = ξ_{0}(ω) = {ξ_{0}^{i}(ω), 1 ≤ i ≤ M}, ψ^{ω}(x) = ψ^{−,ω}(x, 0−) + ψ^{+,ω}(x, 0+), V (x, c) = − ∑_{i=1}^{M} c_{i}F^{i}(x). Let X = {x_{j}, j ∈ ℤ} be a global minimizing sequence. It follows from (93) that x_{0} is a point of minimum for

ψ^{ω}(x) + V (x, c) = ψ^{−,ω}(x, 0−) + ψ^{+,ω}(x, 0+) − ∑_{i=1}^{M} c_{i}F^{i}(x) .

Assumption (35) of Theorem 5 implies that

\frac{∂V}{∂c} (·, c) = −(F^{1}(·), F^{2}(·), ..., F^{M}(·))

is a one-to-one mapping from 𝕋^{d} to ℝ^{M}. Hence, using Corollary 5 we conclude that for Lebesgue-almost all c a point of minimum x_{0} is unique. According to Lemma 9 two global minimizing sequences cannot intersect. This implies that for Lebesgue-almost all c there exists a unique global minimizer. It is easy to see that the set of c for which uniqueness holds depends only on {ξ_{j}, j ≠ 0}. To finish the proof it is enough to notice that the random vector c = ξ_{0}(ω) is independent from {ξ_{j}, j ≠ 0} and has absolutely continuous distribution with respect to the Lebesgue measure.

We next consider the “white force” case. The strategy is similar to the one used in the “kicked” case. However in the “white force” case there are two serious difficulties. The first one is connected with the use of the Rademacher Theorem which is not immediately available in the case of “white” noises. Secondly, the control of the term A_{s,t}^{ω}(γ(s), γ(t)) in Proposition 3 is not as straightforward as in the “kicked” case. To overcome these difficulties we explicitly add drift terms to the the Brownian motions. Namely, for any vector c = (c^{−}, c^{+}) = (c_{i}^{−}, 1 ≤ i ≤ N , c_{i}^{+}, 1 ≤ i ≤ N) ∈ ℝ^{2N} we define

W_{i}^{ω}(t; c, ϵ) = \begin{cases}W_{i}^{ω}(t) + c_{i}^{+}  for t ∈ [ϵ, ∞) \\ W_{i}^{ω}(t) + \frac{c_{i}^{+}}{ϵ} t  for t ∈ [0, ϵ] \\ W_{i}^{ω}(t) + \frac{c_{i}^{−}}{ϵ} t  for t ∈ [−ϵ, 0] \\ W_{i}^{ω}(t) − c_{i}^{−}  for t ∈ (−∞, −ϵ] . \end{cases}  (95)

Denote by T_{c,ϵ} the measurable transformation of the measurable space (Ω, F) corresponding to (95):

T_{c,ϵ} : Ω → Ω, W_{i}^{T_{c,ϵ}(ω)} = W_{i}^{ω}(t; c, ϵ), 1 ≤ i ≤ N .

Let P_{c,ϵ} be the probability measure on (Ω, F) which is the image of measure P under T_{c,ϵ}. Then the following simple consequence of the Cameron-Martin-Girsanov Theorem holds.

Proposition 4. The probability measure P_{c,ϵ} is absolutely continuous with respect to the measure P and the corresponding density is given by

\frac{P_{c,ϵ}(dω)}{P (dω)} = exp ∑_{i=1}^{N} (c_{i}^{+}W_{i}^{ω}(ϵ) − c_{i}^{−}W_{i}^{ω}(−ϵ) − \frac{1}{2}(c_{i}^{+})^{2} − \frac{1}{2}(c_{i}^{−})^{2}) .

The proof of Proposition 4 is standard (see, for example, [41]).

To simplify notations we shall not indicate dependence on ϵ which is everywhere below assumed to be a fixed sufficiently small positive number. We shall denote by A^{ω,c} the action A^{T_{c}(ω)} :

A^{ω,c}(γ) = A_{s,t}^{ω}(γ) − \frac{1}{ϵ} ∑_{k=1}^{N} ∫_{s}^{t} F^{k}(γ(τ))(c_{k}^{−}χ_{−}(τ) + c_{k}^{+}χ_{+}(τ))dτ,  (96)

where A_{s,t}^{ω}(γ) is given by (16) and χ_{−}(τ), χ_{+}(τ) are the indicator functions of the intervals [−ϵ, 0], [0, ϵ] :

χ_{−}(τ) = \begin{cases}1 for τ ∈ [−ϵ, 0] \\ 0 for τ ∈ ℝ − [−ϵ, 0] , \end{cases} χ_{+}(τ) = \begin{cases}1 for τ ∈ [0, ϵ] \\ 0 for τ ∈ ℝ − [0, ϵ] . \end{cases}

Corresponding Lagrangian flow L_{s}^{T_{c}(ω)} is denoted below by L_{s}^{ω,c}. As in (23) we shall say that (x(t), v(t)), t ∈ [t_{1}, t_{2}] is a trajectory of the Lagrangian flow L_{s}^{ω,c} if ẋ(t) = v(t) and for any s ≤ t, s, t ∈ [t_{1}, t_{2}]

v(t) = v(s) + ∫_{s}^{t} ∑_{k=1}^{N}(v(τ) · ∇)∇F^{k}(x(τ))(W_{k}(τ) − W_{k}(s))dτ  (97)

−\frac{1}{ϵ} ∑_{k=1}^{N} ∫_{s}^{t} ∇F^{k}(x(τ))(c_{k}^{−}χ_{−}(τ) + c_{k}^{+}χ_{+}(τ))dτ − ∑_{k=1}^{N} ∇F^{k}(x(t))(W_{k}(t) − W_{k}(s)).

Finally, for any t_{1} < t_{2} we shall denote by Γ_{c,t_{1},t_{2}}^{ω} the set of all minimizers on [t_{1}, t_{2}] for the action A^{ω,c} and by Γ_{c,t}^{ω} the set of all backward minimizers:

Γ_{c,t_{1},t_{2}}^{ω} = {γ_{(x_{1},t_{1};x_{2},t_{2})}^{T_{c}(ω)}, x_{1}, x_{2} ∈ 𝕋^{d}} , Γ_{c,t}^{ω} = {γ_{x,t}^{T_{c}(ω)}, x ∈ 𝕋^{d}} .

For any x, y ∈ 𝕋^{d}, c ∈ ℝ^{N} define

V_{ϵ}^{ω}(x, y, c) = A_{−ϵ,ϵ}^{T_{c}(ω)}(x, y) .

Below we assume that c belongs to

E = {c ∈ ℝ^{2N} : |c_{i}^{−}|, |c_{i}^{+}| ≤ 1, 1 ≤ i ≤ N} .

The idea is that if we add large drift terms \frac{c_{i}^{±}}{ϵ} , 1 ≤ i ≤ N on a small interval of time [−ϵ, ϵ] then V_{ϵ}^{ω}(x, y, c) converges to −δ(x − y) ∑_{k=1}^{N} F^{k}(x)(c_{k}^{−} + c_{k}^{+}) in the limit ϵ → 0. More precisely we have the following proposition.

Proposition 5. For P −almost all ω there exist ϵ_{0} = ϵ_{0}(ω) > 0 and a constant K = K(ω) > 0 such that for all ϵ ≤ ϵ_{0}(ω) and all c ∈ E the following statements hold.

1. V_{ϵ}(x, y, c) is a C^{∞} function on Δ_{ϵ}^{ω} × E, . where

Δ_{ϵ}^{ω} = {(x, y) ∈ 𝕋^{d} × 𝕋^{d} : ρ(x, y) ≤ K(ω)ϵ} .  (98)

2. If γ is a global minimizer for A^{ω,c} then (x = γ(−ϵ), y = γ(ϵ)) belongs to the interior of Δ_{ϵ}^{ω}.

3. The function V_{ϵ}^{ω}(x, y, c) converges to −δ(x − y) ∑_{k=1}^{N} F^{k}(x)(c_{k}^{−} + c_{k}^{+}). Namely,

lim_{ϵ→0} max_{(x,y)∈Δ_{ϵ}^{ω}} max_{c∈E} |V_{ϵ}(x, y, c) + ∑_{k=1}^{N} F^{k}(x)(c_{k}^{−} + c_{k}^{+})| = 0 .  (99)

4. If condition (34) of Theorem 4 holds then the mapping \frac{∂V_{ϵ}}{∂c} (·, ·, c) is an embedding of Δ_{ϵ}^{ω} into ℝ^{2N}.

Proposition 5 is proven in the Appendix. We can now finally prove Theorem 4.

Proof of Theorem 4. Consider the random variable ϵ_{0}(ω). Denote by Ω_{ϵ} = {ω ∈ Ω : ϵ(ω) ≥ ϵ}. Obviously, there exists a constant ϵ̄such that P (Ωϵ̄) ≥ \frac{1}{2}. Fix ϵ = ϵ̄. It follows from Propositions 3, 5 that for any c ∈ E and any global minimizer γ for the action A^{ω,c} a point (γ(−ϵ̄), γ(ϵ̄)) ∈ Δω ; ϵ̄ and corresponds to a minimum of

ψ^{−,ω}(x, −ϵ̄)(x) + ψ^{+,ω}(y, ϵ̄) + Vϵ̄(x, y, c) .  (100)

Since Proposition 5 implies that \frac{∂V_{ϵ}}{∂c} (·, ·, c) is an embedding of Δω ; ϵ̄ we conclude that for P −almost every ω ∈ Ωϵ̄ and Lebesgue-almost every c ∈ E there exists a unique global minimum in (100). Hence, by Lemma 9 there exists a unique global minimizer for the action A^{ω,c}. Fubini Theorem implies that for Lebesgue-almost every c ∈ E there exists a subset of Ωϵ̄ of full P − measure for which uniqueness of global minimizer holds. Let c̄ ∈ E be such a vector. Then there exists a subset Ω_{ϵ̄,c̄} ⊂ Ωϵ̄, P (Ω_{ϵ̄,c̄}) ≥ \frac{1}{2} such that for all ω ∈ Ω_{ϵ̄,c̄} there exists a unique global minimizer for A^{ω,c}. Since A^{ω,c} = A^{T_{c̄,ϵ̄}(ω)} we get the uniqueness of global minimizer for the original action A^{ω} for all ω ∈ T_{c̄,ϵ̄}(Ω_{ϵ̄,c̄}). Notice that

P_{c̄,ϵ̄} (T_{c̄,ϵ̄}(Ω_{ϵ̄,c̄})) = P (Ω_{ϵ̄,c̄}) ≥ \frac{1}{2} .  (101)

Using Proposition 4 we have P (T_{c̄,ϵ̄}(Ω_{ϵ̄,c̄})) > 0. Denote by Ω^{(u)} the set ω ∈ Ω for which the uniqueness of global minimizer for the action A^{ω} holds. It follows from (101) that P (Ω^{(u)}) > 0. Since the set Ω^{(u)} is invariant under time-translations θ^{τ}, τ ∈ ℝ and (Ω, P, θ^{τ}, τ ∈ ℝ) is an ergodic flow we finally get P (Ω^{(u)}) = 1.

We finish with the proof of Corollaries 2, 3 from Theorems 4, 5. Since the proofs are similar in the “white force” and “kicked” cases we consider only the former one. Proof of Corollary 2. Suppose diam(E^{ω,b}(t)) does not tend to 0 as t → ∞. This implies that there exists δ > 0 and an infinite sequence of backward minimizers γ_{x_{n},t_{n}}^{ω,b}(τ) such that t_{n} → ∞ as n → ∞ and

ρ(γ_{x_{n},t_{n}}^{ω,b}(0), x_{g}^{ω,b}) ≥ δ ,

where x_{g}^{ω,b} is the position of the global minimizer at time t = 0. Since velocities γ̇_{x_{n},t_{n}}^{ω,b}(0)of backward minimizers at time t = 0 are bounded there exists a subsequence n_{k} such that (γ_{x_{n_{k}} ,t_{n_{k}}}^{ω,b} (0), γ̇_{x_{n_{k}} ,t_{n_{k}}}^{ω,b} (0)) converges to some (x̄, v̄), where x̄ ∈ 𝕋^{d} and ρ(x̄, x_{g}^{ω,b}) ≥ δ. Then the Lagrangian trajectory γ^{x̄,v̄}(τ), τ ∈ ℝ such that γ^{x̄,v̄}(0) = x̄, γ̇^{x̄,v̄}(0) = v̄ has to be a global minimizer which contradicts to the uniqueness. Proof of Corollary 3. Obviously

L_{s}^{ω} δ_{x_{g}^{ω,b},v_{g}^{ω,b}}(dx, dv) = δ_{x_{g}^{θ^{s}ω,b},v_{g}^{θ^{s}ω,b}}(dx, dv) .

It follows immediately that κ^{b}(dx, dv, dω) is an invariant measure for L̂_{s}, s ∈ ℝ. Notice that under the condition of fixed ω the conditional distributions κ^{b}(dx, dv|ω) = δ_{x_{g}^{ω,b},v_{g}^{ω,b}}(dx, dv) are atomic measures. This together with the ergodicity of P (dω) for

θ^{s}, s ∈ ℝ implies the ergodicity of κ^{b}(dx, dv, dω) for L̂_{s}, s ∈ ℝ.

8. Conclusions and open problems.

We finish the paper with the discussion of the results. We also discuss related problems. Solutions to some of those problems are pretty straightforward and will be published elsewhere in the near future, others remain open. Below we discuss open problems very briefly aiming to convince a reader that they are indeed quite challenging and of significant importance for the future developments of the theory.

1. The results of the paper can be roughly split into two parts. The first group of results are connected with existence and uniqueness of the stationary distributions and related results on uniqueness of solutions to the random Hamilton-Jacobi and Burgers equations for semi-infinite intervals of time (−∞, t_{0}). Those results are of an extremely general nature and are obtained under very mild conditions. The uniqueness of the “viscosity” solutions demonstrates the extreme stability of the random Hamilton-Jacobi and Burgers equations. In short this stability can be formulated as the “one force - one solution” principle. We just mention that in the case of the Navier-Stokes equation situation is much more complicated. The “one force - one solution” principle is valid only for large enough values of the viscosity (see [22, 30, 16, 8]). The second group of results concerns the uniqueness of global minimizer which is a fundamental property of random Lagrangian systems. The conditions here are more restrictive. They are quite satisfactory in the “white force” case where we just need to have enough “pumping” potentials F^{i}(x). In the “kicked” case we have to make a more technical assumption on absolute continuity for the distribution of the random vectors {ξ_{j}^{i}(ω), 1 ≤ i ≤ M}. At the same time one certainly needs to impose some conditions on potentials for the validity of the results. Indeed, if all the potentials have additional symmetry or effectively smaller period (which is in fact also an additional symmetry) then obviously global minimizer is non unique. However this seems to be the only necessary restriction. Unfortunately such general result is quite difficult to prove.

2. As we have already mentioned in the first section Ya. Sinai ([42]) has proved in the beginning of 1990’s the existence and uniqueness of the stationary distribution μ_{ν} for the viscous Burgers equation. It is natural to ask whether μ_{ν} converges weakly to the stationary distribution μ for the inviscid equation. The answer to this question is affirmative. Corresponding result in the one-dimensional case was proven in ([14]). In the multi-dimensional case the proof is based on similar ideas. It will be published in ([28]).

3. It is also possible to show that the invariant measure κ^{b}(dx, dv, dω) for the skewproduct extension L̂_{s} (or L̂_{m} in the “kicked” case) of the Lagrangian flow is in fact the unique action minimizing measure in a sense of Mather. To make this statement precise one should define the skew-product extension of the Lagrangian itself. This can be done in a natural way and the corresponding result will be proved in ([28]).

4. The most important problem which remains open in the theory is the problem of the hyperbolicity of the global minimizer. Since the invariant measure κ^{b}(dx, dv, dω) is ergodic it is possible to define non-random Lyapunov exponents for the Jacobi cocycle: λ_{1} ≥ λ_{2} ≥ ... ≥ λ_{2d}. To prove hyperbolicity one has to show that λ_{i} ≠ 0, 1 ≤ i ≤ 2d. Since the Lagrangian flow is symplectic the Lyapunov exponents appear in pairs of numbers having opposite sign. It follows that in the hyperbolic case there are d stable and d unstable Lyapunov exponents:

λ_{1} ≥ λ_{2} ≥ ... ≥ λ_{d} > 0 > λ_{d+1} ≥ ... ≥ λ_{2d}.

We strongly believe that the following hyperbolicity conjecture holds.

Conjecture. Under assumptions of Theorems 4, 5 the invariant measure κ^{b}(dx, dv, dω) is hyperbolic.

Notice that in the one-dimensional case it is enough to show that λ_{1} > 0. This indeed was proven in ([14]). The proof is based on the exponential estimate for the diam(E(t)). Namely, it was shown that there exist constants α_{1}, α_{2}, K_{1}, K_{2} > 0 such that

P (ω ∈ Ω : diam(E^{ω,b}(t)) ≥ K_{1}e^{α_{1}t}) ≤ K_{2}e^{α_{2}t} .  (102)

Unfortunately the proof in ([14]) is purely one-dimensional. However if the estimate (102) holds in the multi-dimensional case it immediately gives the hyperbolicity conjecture.

5. Hyperbolicity conjecture implies that all “backward” minimizers are asymptotic to the global one as t → −∞. This brings us to an interesting concept of the topological shocks. To explain this notion we have to describe the structure of minimizers on the universal cover ℝ^{d}. Instead of one global minimizer on the torus 𝕋^{d} one has ℤ^{d} lattice of them. Since every “backward” minimizers approaches the global one we get a random tiling of ℝ^{d} onto domains of attraction to a particular global minimizer. The boundaries of these domains correspond to “global” shocks having pure topological nature. Other “local” shocks appear and later merge with the topological singularities. The structure of the topological shocks is quite rigid and crucially depends on the topology of the manifold. One can show that in the case of two-dimensional torus the tiling is formed by a ℤ^{2} lattice of hexagonal domains. In the three-dimensional case the structure of tiling can be more complicated. The notion of topological shock will be discussed in more details in the forthcoming publications. Notice that in the one-dimensional case the topological shock is formed by just a single point. This point was called the main shock in [14]. It is an interesting question to study the structure of topological shocks for general Riemannian manifolds.

6. Another very interesting open problem is related to the random Hamilton-Jacobi and Burgers equation in a non-compact case. It is not known whether the stationary distribution exists even in the case of positive viscosity ν. The only result which we aware of belongs to Yu.Kifer ([29]) who has constructed in the case d ≥ 3 the stationary distribution for the viscous Burgers equation driven by a force which is random in both x and t. We shall also mention recent papers [10, 21] where the non-compact Hamilton-Jacobi equation is studied in the autonomous case.

7. Finally, we shall say few words about physically important quantitative predictions for correlation functions. We hope that in the multi-dimensional case one will be able to obtain non-trivial quantitative results, provided that the hyperbolicity of the global minimizer is established. In the case d = 1 such results were obtained in [15]). It was shown that the pdf g(z) for the velocity gradient z = \frac{du^{ω}}{dx} decays super exponentially for large positive values of z, and algebraically for negative values, i.e. g(z) ∼ \frac{C}{|z|^{α}} as z → −∞. The critical exponent α is universal. It means that α is independent of the precise formula for a driven force. In [40, 6] it was suggested that α is equal to \frac{5}{2} or 3. However, as it was shown in [15] the geometrical picture which follows from the hyperbolicity suggests that the main contribution to g(z) for large negative values of z comes from small neighbourhoods of the preshock events. Then an easy calculation implies that α = \frac{7}{2} (see also [17, 18]). This prediction, although not rigorous, has been verified numerically ([4]) and accepted by the physical community.

Appendix

In this Appendix we prove Proposition 5. The proof of the proposition splits into several lemmas. We assume everywhere that ω ∈ Ω is fixed. To simplify notations we do not indicate dependance on ω although all constants in the estimates below do depend on it. We start with two lemmas which provide estimates for the velocities of minimizers for the action A^{c}.

Lemma 15. For P −almost all ω there exists a constant B > 0 such that for all ϵ ∈ (0, 1] and for all c ∈ E

sup_{γ∈Γ_{c,1}} max_{t∈[−1,1]} |γ̇(t)| ≤ B ,

where Γ_{c,1} is the set of backward minimizers for the action A^{c} starting at time t = 1.

Proof. The proof is similar to the proof of Lemma 6. Since the large drift forces act only on a small time interval we can essentially reproduce the same estimates. For any v_{0} > 0 there exist a constant V (v_{0}) > 0 such that if γ is a trajectory of the Lagrangian flow L_{s}^{c} with c ∈ E, ϵ ∈ (0, 1] and |γ̇(t_{0})| ≥ V (v_{0}) for some t_{0} in [−1, 1] then |γ̇(t) ≥ v_{0} for all t in [−1, 1]. Denote

Ā = sup_{δ(t):|δ̇(t)|≤\sqrt{d}} max_{c∈E} sup_{ϵ∈(0,1]} A_{−1,1}^{c}(δ)

and

A(v_{0}) = inf_{δ(t):|δ̇(t)|≥v_{0}} min_{c∈E} inf_{ϵ∈(0,1]} A_{−1,1}^{c}(δ) .

Clearly, A(v_{0}) → ∞ as v_{0} → ∞. Since any two points x, y ∈ 𝕋^{d} can be connected by the curve δ(t), t ∈ [−1, 1] such that δ̇(t) ≤ \sqrt{d} for all t, it is enough to take B = V (v_{0}), where v_{0} is so large that A(v_{0}) > Ā.

Fix K > 0. Denote Δ_{ϵ}^{K} = {(x, y) ∈ 𝕋^{d} ×𝕋^{d} : ρ(x, y) ≤ Kϵ} and Γ_{c,−ϵ,ϵ}^{K} = {γ ∈ Γ_{c,−ϵ,ϵ} : (γ(−ϵ), γ(ϵ)) ∈ Δ_{ϵ}^{K}}. Then the following lemma holds.

Lemma 16. For a fixed K > 0 and for P −almost all ω ∈ Ω there exists a constant D(K) > 0 such that for all ϵ ∈ (0, 1] and for all c ∈ E

sup_{γ∈Γ_{c,−ϵ,ϵ}^{K}} max_{t∈[−ϵ,ϵ]} |γ̇(t)| ≤ D(K) .

Proof. The proof follows the same lines. However one needs more estimates in order to prove uniformity in ϵ. As above for any v_{0} > 0 there exist a V (v_{0}) > 0 independent of c and ϵ such that if γ is a trajectory of the Lagrangian flow L_{s}^{c} with c ∈ E, ϵ ∈ (0, 1] and |γ̇(t_{0})| ≥ V (v_{0}) for some t_{0} in [−ϵ, ϵ] then |γ̇(t)| ≥ v_{0} for all t in [−ϵ, ϵ]. Denote

Ā(K) =
sup_{δ(t):|δ̇(t)|≤K/2} max_{c∈E} sup_{ϵ∈(0,1]} \frac{1}{ϵ} (A_{−ϵ,ϵ}^{c}(δ) + ∑_{k=1}^{N} [(c_{k}^{−} + c_{k}^{+} + W_{k}(ϵ) − W_{k}(−ϵ))F^{k}(δ(ϵ))])

and

A_{1}(v_{0}) =

inf_{δ(t):|δ̇(t)|≥v_{0}} min_{c∈E} inf_{ϵ∈(0,1]} \frac{1}{ϵ} (A_{−ϵ,ϵ}^{c}(δ) + ∑_{k=1}^{N} [(c_{k}^{−} + c_{k}^{+} + W_{k}(ϵ) − W_{k}(−ϵ))F^{k}(δ(ϵ))]) .

It follows from (16), (96) that for any K ≥ 0 there exists a constant C(K) such that

|A_{−ϵ,ϵ}^{c}(δ) + ∑_{k=1}^{N} [(c_{k}^{−} + c_{k}^{+} + W_{k}(ϵ) − W_{k}(−ϵ))F^{k}(δ(ϵ))]| ≤ C(K)ϵ ,

provided |δ̇(t)| ≤ K/2 for all t ∈ [−ϵ, ϵ]. Hence, Ā(K) is finite. We next show that A_{1}(v_{0}) is also finite. Taking a curve δ_{0}(t) such that |δ̇_{0}(t)| = v_{0} for all t ∈ [−ϵ, ϵ] we obtain

A_{1}(v_{0}) ≤ \frac{1}{ϵ} (A_{−ϵ,ϵ}^{c}(δ) + ∑_{k=1}^{N} [(c_{k}^{−} + c_{k}^{+} + W_{k}(ϵ) − W_{k}(−ϵ))F^{k}(δ(ϵ))]) ≤ C_{0}

for some constant C_{0} > 0. Denote

I_{0} = ∫_{−ϵ}^{ϵ} \frac{1}{2}(δ̇(τ) − b)^{2}dτ , I_{1} = ∫_{−ϵ}^{ϵ} (∑_{k=1}^{N} ∇F^{k}(δ(τ)) · δ̇(τ)(W_{k}(τ) − W_{k}(−ϵ)) − \frac{b^{2}}{2} ) dτ ,

I_{2} = \frac{1}{ϵ} ∑_{k=1}^{N} ∫_{−ϵ}^{ϵ}(F^{k}(δ(ϵ)) − F^{k}(δ(τ)))(c_{k}^{−}χ_{−}(τ) + c_{k}^{+}χ_{+}(τ))dτ .

Then

A_{−ϵ,ϵ}^{c}(δ) + ∑_{k=1}^{N} [(c_{k}^{−} + c_{k}^{+} + W_{k}(ϵ) − W_{k}(−ϵ))F^{k}(δ(ϵ))] = I_{0} + I_{1} + I_{2} .

It is easy to show that there exists a constant C_{1} > 0 such that

|I_{1}| , |I_{2}| ≤ C_{1}(ϵ + \sqrt{I_{0}}\sqrt{ϵ}) .

Hence,

I_{0} + I_{1} + I_{2} ≥ I_{0} − 2C_{1}\sqrt{I_{0}}\sqrt{ϵ} − 2C_{1}ϵ ≥ −(C_{1}^{2} + 2C_{1})ϵ .  (103)

This implies that A_{1}(v_{0}) is finite. Moreover, it follows from (103) that A_{1}(v_{0}) → ∞ as v_{0} → ∞. Take now v_{0} so large that A_{1}(v_{0}) > Ā(K). It is easy to see the statement of the lemma holds for D(K) = V (v_{0}).

The following lemma shows that two Lagrangian trajectories which start at the same point at time t = −ϵ do not intersect till time ϵ.

Lemma 17. For a fixed D > 0 and for P −almost all ω ∈ Ω there exist a constant ϵ_{0}(D) > 0 such that for any two trajectories δ^{(1)}(t), δ^{(2)}(t) of the Lagrangian flow L_{s}^{c} with c ∈ E, ϵ ∈ (0, ϵ_{0}(D)] we have

ρ(δ^{(1)}(t), δ^{(2)}(t)) ≥ \frac{1}{2\sqrt{d}}(t + ϵ)|δ̇^{(1)}(−ϵ) − δ̇^{(2)}(−ϵ)| , t ∈ [−ϵ, ϵ] ,  (104)

provided δ^{(1)}(t), δ^{(2)}(t) satisfy conditions

δ^{(1)}(−ϵ) = δ^{(2)}(−ϵ), |δ̇^{(1)}(−ϵ)|, |δ̇^{(2)}(−ϵ)| ≤ D .  (105)

Proof. Let δ^{(1)}(t), δ^{(2)}(t) be the trajectories of the Lagrangian flow L^{c} which satisfy (105). Denote

δ̇^{(1),(2)}(t) = v^{(1),(2)}(t), ρ(t) = ρ(δ^{(1)}(t), δ^{(2)}(t)), b(t) = |v^{(1)}(t) − v^{(2)}(t)| .

Using (97) it is easy to show that there exists a constant D_{1} such that |v^{(1),(2)}(t)| ≤ D_{1} for all t ∈ [−ϵ, ϵ]. Appling this estimate and again (97) we have

b(t) ≤ b(−ϵ) + K_{1} (∫_{−ϵ}^{t} b(τ)dτ + \frac{1}{ϵ} ∫_{−ϵ}^{t} ∫_{−ϵ}^{τ} b(s)ds dτ)

for some constant K_{1} > 0. It follows that there exists a constant K_{2} > 0 such that

b(t) ≤ b(−ϵ) + K_{2} ∫_{−ϵ}^{t} b(τ)dτ .

Hence, by the Gronwall inequality

b(t) ≤ b(−ϵ)e^{K_{2}(t+ϵ)} ,  (106)

which gives an estimate

ρ(t) ≤ K_{3}ϵb(−ϵ), t ∈ [−ϵ, ϵ] ,  (107)

for ϵ ∈ (0, 1] and a constant K_{3} > 0. Using once again (97) and the estimates (106), (107) we get

|(v^{(1)}(t) − v^{(2)}(t)) − (v^{(1)}(−ϵ) − v^{(2)}(−ϵ))| ≤ K_{4}ϵb(−ϵ) ,

where K_{4} is a positive constant. To complete the proof choose a component of velocity such that

|v_{i}^{(1)}(−ϵ) − v_{i}^{(2)}(−ϵ))| ≥ \frac{1}{\sqrt{d}}b(−ϵ), 1 ≤ i ≤ d .

Then for all t ∈ [−ϵ, ϵ]

|v_{i}^{(1)}(t) − v_{i}^{(2)}(t))| ≥ (\frac{1}{\sqrt{d}} − K_{4}ϵ) b(−ϵ) .  (108)

Take ϵ_{0}(D) so small that

4Dϵ_{0}(D)e^{2K_{2}ϵ_{0}(D)} ≤ \frac{1}{2} , K_{4}ϵ_{0}(D) ≤ \frac{1}{2\sqrt{d}} .

Then using (106) and (108) we finally obtain

\frac{1}{2\sqrt{d}}(t + ϵ)b(−ϵ) ≤ |δ_{i}^{(1)}(t) − δ_{i}^{(2)}(t))| ≤ \frac{1}{2} , t ∈ [−ϵ, ϵ] ,

which implies ρ(t) ≥ \frac{1}{2\sqrt{d}}(t + ϵ)b(−ϵ).

Remark 8.1. All estimates in Lemmas 15, 16 and 17 were proven for c ∈ E. In the same way one can prove all three lemmas for c ∈ E_{a} = {c ∈ ℝ^{2N} : |c_{i}^{−}|, |c_{i}^{+}| ≤ a, 1 ≤ i ≤ N}. We shall denote corresponding constants B(a), D(K, a), ϵ_{0}(D, a) .

Let B be the bound given by Lemma 15. Define K = 2B and Δ_{ϵ} = {(x, y) ∈ 𝕋^{d} × 𝕋^{d} : ρ(x, y) ≤ Kϵ}. To formulate the next lemma we shall also need constants D = D(K + 1, 2) and and ϵ_{0} = ϵ_{0}(D, 2) which are defined in Remark 8.1.

Lemma 18. For all ϵ ∈ (0, ϵ_{0}] and all c ∈ E the following statements hold. 1. If γ(τ), τ ∈ (−∞, 1] belongs to Γ_{c,1} then (γ(−ϵ), γ(ϵ)) ∈ Δ_{ϵ}.

2. For every (x, y) ∈ Δ_{ϵ} there exists a unique γ(τ), τ ∈ [−ϵ, ϵ] which minimizes the action A_{s,t}^{c}γ) with fixed end-points x = γ(−ϵ), y = γ(ϵ).

3. The function V_{ϵ}(x, y, c) is C^{∞}(Δ_{ϵ} × E).

Proof. The proof of the first statement follows directly from Lemma 15 and the choice of the constant K. To prove the second statement notice that according to Lemma 16 all minimizers will have velocities with the norm bounded by D. Since all minimizers are Lagrangian trajectories it follows from Lemma 17 that for all ϵ ∈ (0, ϵ_{0}] and all (x, y, c) in a neighbourhood of Δ_{ϵ} × E there exists a unique minimizer. Finally, we shall show that the differentiability of V_{ϵ}(x, y, c) is a consequence of the uniqueness of minimizers. Fix arbitary ϵ ∈ (0, ϵ_{0}]. Let γ_{x,v} be a trajectory of the Lagrangian flow L^{c} with x = γ(−ϵ), v = γ̇(−ϵ). Denote y = γ(ϵ), w = γ̇(ϵ). Then the action A_{−ϵ,ϵ}^{c}(γ_{x,v}) and the Lagrangian flow L_{−ϵ,ϵ}^{c} : (x, v) ↦ (y, w) are C^{∞} functions of the variables (x, v) ∈ 𝕋^{d}×ℝ^{d} and c ∈ ℝ^{2N}. Lemma 17 implies that for all (x, y, c) in a neighbourhood of Δ_{ϵ}×E there exists a unique v = v(x, y, c) such that L_{−ϵ,ϵ}^{c} : (x, v) ↦ (y, ·) and |v| ≤ D. It follows from (104) that \frac{∂y}{∂v}(x, v, c) is a nondegenerate matrix for all x ∈ 𝕋^{d}, |v| ≤ D.

Using the Implicit Function Theorem we conclude that v = v(x, y, c) and, hence, V_{ϵ}(x, y, c) = A_{−ϵ,ϵ}^{c}(γ_{x,v}) are C^{∞} functions on Δ_{ϵ} × E.

In the next two lemmas we study the partial derivatives of the function V_{ϵ}(x, y, c).

Lemma 19.

1. The following asymptotic relations hold:

lim_{ϵ→0} max_{(x,y)∈Δ_{ϵ}} max_{c∈E} |\frac{∂^{2}V_{ϵ}}{∂x∂c_{i}^{−}} (x, y, c) + \frac{3}{4}∇F^{i}(x)| = 0 ,

lim_{ϵ→0} max_{(x,y)∈Δ_{ϵ}} max_{c∈E} |\frac{∂^{2}V_{ϵ}}{∂y∂c_{i}^{−}} (x, y, c) + \frac{1}{4}∇F^{i}(x)| = 0 ,  (109)

lim_{ϵ→0} max_{(x,y)∈Δ_{ϵ}} max_{c∈E} |\frac{∂^{2}V_{ϵ}}{∂x∂c_{i}^{+}} (x, y, c) + \frac{1}{4}∇F^{i}(x)| = 0 ,

lim_{ϵ→0} max_{(x,y)∈Δ_{ϵ}} max_{c∈E} |\frac{∂^{2}V_{ϵ}}{∂y∂c_{i}^{+}} (x, y, c) + \frac{3}{4}∇F^{i}(x)| = 0 .

2. There exists a constant C_{V} > 0 such that for all ϵ ∈ (0, 1]

max_{(x,y)∈Δ_{ϵ}} max_{c∈E} ∥\frac{∂^{3}V_{ϵ}}{∂^{2}(x, y)∂c}(x, y, c)∥ ≤ C_{V} .  (110)

Proof. We shall prove only the first statement since the proof of the second statement follows the same path. For a fixed c ∈ E, (x_{0}, y_{0}) ∈ Δ_{ϵ} denote γ(τ; x_{0}, y_{0}, c) = γ(τ), τ ∈ [−ϵ, ϵ] a unique trajectory of the Lagrangian flow L_{s}^{c} connecting points (x_{0}, −ϵ) and (y_{0}, ϵ). Let v(τ; x_{0}, y_{0}, c) = v(τ) = γ̇(τ; x_{0}, y_{0}, c), τ ∈ [−ϵ, ϵ] be the velocity of γ(τ). Since V_{ϵ}(x, y, c) is a smooth function on Δ_{ϵ} × E it follows from Lemma 10 that

\frac{∂V_{ϵ}}{∂y} (x_{0}, y_{0}, c) = v(ϵ; x_{0}, y_{0}, c) − b .

Changing t to −t we also get

\frac{∂V_{ϵ}}{∂x} (x_{0}, y_{0}, c) = −v(−ϵ; x_{0}, y_{0}, c) + b .

Hence,

\frac{∂^{2}V_{ϵ}}{∂y∂c}(x_{0}, y_{0}, c) = \frac{∂v_{x_{0},y_{0},c}}{∂c} (ϵ), \frac{∂^{2}V_{ϵ}}{∂x∂c}(x_{0}, y_{0}, c) = −\frac{∂v_{x_{0},y_{0},c}}{∂c} (−ϵ) .  (111)

Denote

γ_{c}(τ) = \frac{∂γ}{∂c} (τ; x_{0}, y_{0}, c), v_{c}(τ) = \frac{∂v}{∂c} (τ; x_{0}, y_{0}, c) = γ̇_{c}(τ) .

Differentiating with respect to c in equation (97) we get the following equation:

v_{c}(t) = v_{c}(−ϵ) + ∫_{−ϵ}^{t} ∑_{k=1}^{N}(v_{c}(τ) · ∇)∇F^{k}(γ(τ))(W_{k}(τ) − W_{k}(s))dτ
+ ∫_{−ϵ}^{t} ∑_{k=1}^{N}(v(τ) · (γ_{c}(τ) · ∇)∇)∇F^{k}(γ(τ))(W_{k}(τ) − W_{k}(s))dτ  (112)
− ∫_{−ϵ}^{t} \frac{1}{ϵ} ∑_{k=1}^{N}(γ_{c}(τ) · ∇)∇F^{k}(γ(τ))a_{k}(τ, c)dτ − ∫_{−ϵ}^{t} \frac{1}{ϵ} ∑_{k=1}^{N} ∇F^{k}(γ(τ))\frac{∂a_{k}}{∂c} (τ, c)dτ
− ∑_{k=1}^{N}(γ_{c}(t) · ∇)∇F^{k}(γ(t))(W_{k}(t) − W_{k}(s)) ,

where

a_{k}(τ, c) = \begin{cases}c_{k}^{−} for τ ∈ [−ϵ, 0) \\ c_{k}^{+} for τ ∈ (0, ϵ] . \end{cases} (113)

It follows from (113) that \frac{∂a_{k}}{∂c_{i}^{−,+}} (τ, c) = 0 for k ≠ i and

\frac{∂a_{k}}{∂c_{k}^{−}} (τ, c) = \begin{cases}1 for τ ∈ [−ϵ, 0) \\ 0 for τ ∈ (0, ϵ] , \end{cases} \frac{∂a_{k}}{∂c_{k}^{+}} (τ, c) = \begin{cases}0 for τ ∈ [−ϵ, 0) \\ 1 for τ ∈ (0, ϵ] . \end{cases}  (114)

Using obvious boundary conditions γ_{c}(−ϵ) = 0, γ_{c}(ϵ) = 0 we also have

|γ_{c}(t)| ≤ t max_{τ∈[−ϵ,t]} |v_{c}(τ)| , ∫_{−ϵ}^{ϵ} v_{c}(t)dt = 0 .  (115)

Define

ṽ_{c}(t) = v_{c}(−ϵ) − ∑_{k=1}^{N} ∇F^{k}(x_{0}) ∫_{−ϵ}^{t} \frac{1}{ϵ} \frac{∂a_{k}}{∂c} (τ, c)dτ , w(t) = max_{τ∈[−ϵ,t]} |v_{c}(τ) − ṽ_{c}(τ)| .

Clearly there exists a constant K > 0 such that

max_{t∈[−ϵ,ϵ]} |ṽ_{c}(t| ≤ K + |v_{c}(−ϵ)| .  (116)

Using Lemma 16 and (112), (116) it is easy to show that there exist constants K_{1}, K_{2} > 0 such that for all ϵ small enough

w(t) ≤ K_{1}ϵ(K + |v_{c}(−ϵ)|) + K_{2} ∫_{−ϵ}^{t} w(τ)d(τ) .

Hence, using Gronwall inequality we have

w(t) ≤ K_{1}ϵ(K + |v_{c}(−ϵ)|)e^{K_{2}ϵ}, t ∈ [−ϵ, ϵ] .  (117)

Below we denote v_{c}(−ϵ) by v_{c}. Notice that

∫_{−ϵ}^{ϵ} ṽ_{c_{i}^{−}}(t)dt = 2ϵv_{c_{i}^{−}} − ∇F^{i}(x_{0}) (∫_{−ϵ}^{0} \frac{t + ϵ}{ϵ}dt + ∫_{0}^{ϵ} dt) = 2ϵ (v_{c_{i}^{−}} − \frac{3}{4}∇F^{i}(x_{0})) (118)

and

∫_{−ϵ}^{ϵ} ṽ_{c_{i}^{+}}(t)dt = 2ϵv_{c_{i}^{+}} − ∇F^{i}(x_{0}) ∫_{0}^{ϵ} \frac{t}{ϵ}dt = 2ϵ (v_{c_{i}^{+}} − \frac{1}{4}∇F^{i}(x_{0})) .  (119)

Using (115), (117) - (119) we obtain

|v_{c_{i}^{−}} − \frac{3}{4}∇F^{i}(x_{0})| , |v_{c_{i}^{+}} − \frac{1}{4}∇F^{i}(x_{0})| ≤ K_{1}ϵ(K + |v_{c}|)e^{K_{2}ϵ} , 1 ≤ i ≤ N .

It follows that for ϵ small enough

max_{1≤i≤N}(|v_{c_{i}^{−}}| , |v_{c_{i}^{+}}|) ≤ M = 1 + max_{1≤i≤N} max_{x_{0}∈𝕋^{d}} |∇F^{i}(x_{0})| ,  (120)

which implies

|v_{c_{i}^{−}} − \frac{3}{4}∇F^{i}(x_{0})| , |v_{c_{i}^{+}} − \frac{1}{4}∇F^{i}(x_{0})| ≤ K_{3}ϵ , 1 ≤ i ≤ N  (121)

for some constant K_{3} > 0. We next estimate v_{c}(ϵ). Since

ṽ_{c_{i}^{−}}(ϵ) = v_{c_{i}^{−,+}} − ∇F^{i}(x_{0})

we finally get from (117), (120), (121)

|v_{c_{i}^{−}}(ϵ) + \frac{1}{4}∇F^{i}(x_{0})| , |v_{c_{i}^{+}}(ϵ) + \frac{3}{4}∇F^{i}(x_{0})| ≤ 2K_{3}ϵ , 1 ≤ i ≤ N .  (122)

Since estimates (121), (122) are uniform in (x, y) ∈ Δ_{ϵ} and c ∈ E they together with (111) immediately imply (109).

Lemma 20.

lim_{ϵ→0} max_{(x,y)∈Δ_{ϵ}} max_{c∈E} |\frac{∂V_{ϵ}}{∂c_{i}^{−,+}} (x, y, c) − F^{i}(x)| = 0 .  (123)

Proof. We shall use the same notations as in the proof of Lemma 19. It follows from the proof of the previous lemma that there exists a constant K_{4} > 0 such that

max_{t∈[−ϵ,ϵ]} |v_{c}(t)| ≤ K_{4} , max_{t∈[−ϵ,ϵ]} |γ_{c}(t)| ≤ K_{4}ϵ .  (124)

Differentiating with respect to c in (96) and using (124) it is easy to show that there exists a constant K_{5} > 0 such that

|\frac{∂V_{ϵ}}{∂c_{i}^{−,+}} (x, y, c) − F^{i}(x)| ≤ K_{5}ϵ ,

which immediately implies the statement of lemma.

Proof of Proposition 5. The first two statements were proven in Lemma 18. The third statement follows easily from Lemma 16 and (16), (96). We finish with the proof of the last statement. Since the mapping (F^{1}(x), ... , F^{N}(x)) : 𝕋^{d} → ℝ^{N} is an embedding, the matrices \frac{∂F^{i}}{∂x_{j}} have rank d. Hence, there exists an absolute constant C_{1} > 0 such that for any vector r ∈ ℝ^{d}

max_{1≤j≤d} |r · ∇F^{j}(x)| ≥ C_{1}|r| .  (125)

Let (x_{1}, y_{1}), (x_{2}, y_{2}) ∈ Δ_{ϵ}. Define r_{x} = x_{2} − x_{1}, r_{y} = y_{2} − y_{1},

A = \frac{∂V_{ϵ}}{∂c} (x_{2}, y_{2}) − \frac{∂V_{ϵ}}{∂c} (x_{1}, y_{1})

and

|A| = max_{1≤i≤N} max (|\frac{∂V_{ϵ}}{∂c_{i}^{−}} (x_{2}, y_{2}) − \frac{∂V_{ϵ}}{∂c_{i}^{−}} (x_{1}, y_{1})| , |\frac{∂V_{ϵ}}{∂c_{i}^{+}} (x_{2}, y_{2}) − \frac{∂V_{ϵ}}{∂c_{i}^{+}} (x_{1}, y_{1})|) .

It follows from Lemma 19 that there exists a constant C_{2} > 0 such that

A = \frac{∂^{2}V_{ϵ}}{∂(x, y)∂c}(x_{1}, y_{1})(r_{x}, r_{y}) + Q,

where |Q| ≤ C_{2}(|r_{x}| + |r_{y}|)^{2}. Lemma 19 also implies that for ϵ small enough

|\frac{∂^{2}V_{ϵ}}{∂(x, y)∂c_{i}^{−}} (x_{1}, y_{1})(r_{x}, r_{y}) + \frac{3}{4}(r_{x} · ∇F^{i}(x)) + \frac{1}{4}(r_{y} · ∇F^{i}(x))| ≤ \frac{C_{1}}{16} max(|r_{x}|, |r_{y}|) ,

|\frac{∂^{2}V_{ϵ}}{∂(x, y)∂c_{i}^{−}} (x_{1}, y_{1})(r_{x}, r_{y}) + \frac{3}{4}(r_{x} · ∇F^{i}(x)) + \frac{1}{4}(r_{y} · ∇F^{i}(x))| ≤ \frac{C_{1}}{16} max(|r_{x}|, |r_{y}|) .

Denote

M_{i} = max (|\frac{3}{4}(r_{x} · ∇F^{i}(x)) + \frac{1}{4}(r_{y} · ∇F^{i}(x))| , |\frac{1}{4}(r_{x} · ∇F^{i}(x)) + \frac{3}{4}(r_{y} · ∇F^{i}(x))|) .

It is easy to see that

M_{i} ≥ \frac{1}{8} max(|r_{x} · ∇F^{i}(x)| , |r_{y} · ∇F^{i}(x)|) .  (126)

It follows from (125) that there exists 1 ≤ i ≤ N such that M_{i} ≥ \frac{C_{1}}{8} max(|r_{x}|, |r_{y}|). Using the estimates above we have

|A| ≥ max_{1≤i≤N} M_{i} − \frac{C_{1}}{16} max(|r_{x}|, |r_{y}|) − C_{2}(|r_{x}| + |r_{y}|)^{2}
≥ \frac{C_{1}}{16} max(|r_{x}|, |r_{y}|) − C_{2}(|r_{x}| + |r_{y}|)^{2} > 0 ,

provided max(|r_{x}|, |r_{y}|) < m = \frac{C_{1}}{64C_{2}} . Consider now the case when max(|r_{x}|, |r_{y}|) ≥ m. Since (x_{1}, y_{1}), (x_{2}, y_{2}) ∈ Δ_{ϵ} it follows that |r_{x}| ≥ m/2 if ϵ is small enough. Using again that the mapping (F^{1}(x), ... , F^{N}(x)) : 𝕋^{d} → ℝ^{N} is an embedding we conclude that

ΔF = max_{x_{1},x_{2}∈𝕋^{d}:|x_{1}−x_{2}|≥m/2} max_{1≤i≤N} |F^{i}(x_{1}) − F^{i}(x_{2})| > 0 .

This together with Lemma 20 imply that for ϵ small enough and for all (x_{1}, y_{1}), (x_{2}, y_{2}) ∈ Δ_{ϵ} such that max(|r_{x}|, |r_{y}|) ≥ m we have

\frac{∂V_{ϵ}}{∂c} (x_{1}, y_{1}) ≠ \frac{∂V_{ϵ}}{∂c} (x_{2}, y_{2}) .  (127)

It follows from (127), (127) that for ϵ small enough the mapping \frac{∂V_{ϵ}}{∂c} (·, ·, c) is one-to-one on Δ_{ϵ}^{ω}. The estimate (126) and Lemma 19 also imply that for ϵ small enough and for all (x, y) ∈ Δ_{ϵ} the matrices \frac{∂^{2}V_{ϵ}}{∂(x,y)∂c}(x, y, c) have maximum rank 2d. Hence, the mapping \frac{∂V_{ϵ}}{∂c} (·, ·, c) is an embedding.

Acknowledgement. We are grateful to J.Bec, A.Fahti, U. Frisch, G.Contreras, P.Plotnikov, V.Sedykh and Ya.Sinai for helpful discussions. The work was supported by EPSRC research grant GR/M45610, with further support for R. Iturriaga from Conacyt grant 28489-E.

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R. Iturriaga: Centro de Investigacion en Matematica, Apartado Postal 402, Guanajuato Gto., 36000 Mexico; Isaac Newton Institute for Mathematical Sciences, University of Cambridge.

  E-mail address: renato@cimat.mx

K.Khanin: Isaac Newton Institute for Mathematical Sciences, University of Cambridge, 20 Clarkson Road, Cambridge CB3 0EH, UK; BRIMS, Hewlett-Packard Laboratories, Bristol; Department of Mathematics, Heriot-Watt University, Edinburgh; Landau Institute for Theoretical Physics, Moscow.

  E-mail address: K.Khanin@newton.cam.ac.uk