Large Deviations for Dissipative Diffusion Equations in Conuclear

Spaces

Víctor Pérez-Abreu CIMAT Apdo Postal 402, Guanajuato, 36000, Gto. México and Constantin Tudor University of Bucharest and CIMAT Department of Mathematics 70109 Bucharest, Romania

Abstract

This paper uses a general extended contraction principle due to the authors, to prove large deviations for dissipative diffusion equations in duals of nuclear Fréchet spaces.

Key Words: Countably hilbertian nuclear space, Diffusion equations, Large deviations. AMS Classification: Primary 60H10, Secondary 60J60.

1 Introduction

Large deviations for autonomous finite dimensional nonanticipating Itô equations were originally studied by Freidlin and Wentzell (see [9]). Extensions to more general equations in finite dimensions are given by Azencot [1], Baldi and Chaleyat-Maurel [2], Cutland [4], Dozzi [10], Liptser and Pukhalskii [14] and Priouret [18]. The infinite dimensional case is considered by Peszat [17] in the Hilbert space situation, and by Pérez-Abreu and Tudor [16] and Xiong [21] for nuclear space valued stochastic equations. In all these papers the time parameter space has a first finite element. However, there are situations where the time parameter is the whole real axis, as shown, for example, in the works by Da Prato and Tudor [5] and Tudor [19]. In this paper we use a general useful criterion for proving large deviations due to Pérez-Abreu and Tudor [15], to prove this principle for dissipative nuclearvalued diffusion processes with the whole real line as time parameter. The finite dimensional case is considered in Tudor [20]. The basic assumptions on the coefficients of the diffusion are similar to those in Kallianpur and Xiong [12], but they are global in the sense that are independent of time. It is worthwhile to mention that the assumptions are formulated in a nuclear set-up, involving the whole sequence of Hilbert spaces which defines the nuclear space, and therefore a single Hilbert space structure is not appropriate.

2 Notation and Preliminaries

2.1 Large Deviations

In this subsection we review some useful facts about large deviations which are used in the sequel. Let (S, d) be a Polish space, {μ_{n}}_{n≥1} be a sequence of probability measures on the Borel σ-algebra of S and Λ : S → [0, +∞] be a function. We recall that the sequence {μ_{n}}_{n≥1} is said to satisfy the large deviations principle (LDP) with the rate function Λ if :

(a) Λ is lower semicontinuous.

(b) For every nonempty open set G ⊂ S

−Λ(G) ≤\underline{lim}_{n→∞} n^{−1} log μ_{n}(G).

(c) For every closed set F ⊂ S

\overline{lim_{n→∞}} n^{−1} log μ_{n}(F) ≤ −Λ(F),

where, for a set A ⊂ S, Λ(A) = inf_{x∈A} Λ(x).

We say that the rate function Λ is good if the level set Ψ_{Λ}(r) = {x ∈ S : Λ(x) ≤ r} is compact for each r ≥ 0.

A useful criterion for large deviations based on “approximating sequences of LDP” is the following result due to Pérez-Abreu and Tudor [15]:

Theorem 2.1 Let (S′, d′) be a Polish space, (Ω, F, P) be a probability space and Λ : S → [0, ∞] be a good rate function with the skeleton Ψ_{Λ} = ⋃_{r>0} Ψ_{Λ}(r).

Let H_{k}, H : Ψ_{Λ} → S′ be transformations and X_{k}^{n}, X^{n} : Ω → S′ (n, k ≥ 1) be random elements. Assume that :

(1) For every r ≥ 0, H_{k} |_{Ψ_{Λ}(r)} ⟶_{k→∞} H |_{Ψ_{Λ}(r)} uniformly, and H_{k} |_{Ψ_{Λ}(r)}are continuous for all k.

(2) For every k, the sequence {X_{k}^{n}}_{n≥1} satisfies the LDP with the good rate function

Λ′_{k}(y) = inf {Λ(x) : x ∈ Ψ_{Λ}, H_{k}(x) = y} .  (2.1)

(3) The family {X_{k}^{n}}_{k,n≥1} is an exponentially good approximation of {X^{n}}_{n≥1}, i.e.,

lim_{k→∞} \overline{lim}_{n→∞} n^{−1} log P (d′(X_{k}^{n}, X^{n}) > δ) = −∞, ∀δ > 0.  (2.2)

Then the sequence {X^{n}}_{n≥1} satisfies the LDP on S′ with the good rate function

Λ′(y) = inf {Λ(x) : x ∈ Ψ_{Λ}, H(x) = y} .  (2.3)

When proving LDP’s for sequences on product spaces the following result due to Chaganty [3] is a useful technique. Theorem 2.2 Let (S′, d′) be a Polish space and let {μ_{n}}_{n≥1} be a sequence of probability measures on S which satisfies the LDP with the good rate function Λ. Let {ν_{n}(x, dy)}_{n≥1} be a sequence of transition probability measures on S × S′ such that, for every {x_{n}}_{n≥1} ⊂ S with x_{n} → x, the sequence {ν_{n}(x_{n}, dy)}_{n≥1} satisfies the LDP on S′ with the good rate function J(x, .), where the function J : S × S′ → [0, ∞] is lower semicontinuous. Then the sequence {μ_{n} ⊗ ν_{n}}_{n≥1} satisfies the LDP on S × S′ with the good rate function

I(x, x′) = J(x, x′) + Λ(x).  (2.4)

2.2 Nuclear-valued SDEs

In this section we present known results and facts on nuclear space valued processes. Let be Φ a countably Hilbertian nuclear space with the strong dual Φ′. Let {∥ · ∥_{p}}_{p≥0} be an increasing sequence of Hilbertian seminorms defining the topology of Φ. Let Φ_{p} be the completion of Φ by ∥ · ∥_{p} and let Φ_{−p} be the dual of Φ_{p} with the dual norm ∥ · ∥_{−p}. Recall that Φ = ∩_{p}Φ_{p}, Φ′ = ∪_{p}Φ_{−p} and for each p there exists a q ≥ p such that the canonical injection i_{qp} : Φ_{q} → Φ_{p} is Hilbert–Schmidt, i.e. if {e_{j}^{q}}_{j≥1} is a complete orthonormal system (CONS) in Φ_{q} then ∑_{j=1}^{∞} ∥ e_{j}^{q} ∥_{p}^{2}< ∞. For {φ_{j}^{p}}_{j≥1} a CONS in Φ_{p} , we denote by {φ_{j}^{−p}}_{j≥1} the corresponding con-

jugate CONS in Φ_{−p} and by θ_{p} : Φ_{−p} → Φ_{p} the isometry θ_{p}φ_{j}^{−p} = φ_{j}^{p}, for every j ≥ 1. Let Q : Φ × Φ → R be a covariance functional, i.e. a positive definite continuous symmetric bilinear form on Φ × Φ. Let {W_{t}^{(i)}}_{t≥0}, i = 1, 2, be two independent Q–Wiener processes with values in Φ′. Define the Φ′–valued process {W_{t}}_{t∈R} by

W_{t} = \{ \begin{matrix}W_{t}^{(1)} & if t ≥ 0 \\ W_{−t}^{(2)} & if t ≤ 0 \end{matrix}

and for each t ∈ R let F_{t} = B(W_{u} − W_{v} : −∞ < u ≤ v ≤ t). We call W a Q–Wiener process with time parameter R. Let H_{Q} ⊂ Φ′ be the Hilbert space obtained as the completion of the linear subspace

R(i) = {Q(φ, .) : φ ∈ Φ} ⊂ Φ′,

with respect the scalar product

⟨Q(φ_{1}, .), Q(φ_{2}, .)⟩_{H_{Q}} = Q(φ_{1}, φ_{2}),

(the application i : Φ → Φ′, iφ = Q(φ, .) is injective). The space H_{Q} is called the reproducing kernel Hilbert space generated by Q. Define the Cameron-Martin (Hilbert) spaces as

H^{T} = {g(t) = ∫_{0}^{t} ġ(s)ds, t ∈ [0, T] , ġ ∈ L^{2}([0, T] , H_{Q})} , T > 0,

H = {g(t) = ∫_{0}^{t} ġ(s)ds, t ∈ R, ġ ∈ L^{2}(R, H_{Q})} ,

∥g∥_{H^{T}} = \left∥{ \atop ġ}\right∥_{L^{2}([0,T],H_{Q})} , g ∈ H^{T} ,

∥g∥_{H} = \left∥{ \atop ġ}\right∥_{L^{2}(R,H_{Q})} , g ∈ H.

Remark 2.3 It holds that B ∈ L_{2}(H_{Q}, Φ_{−p}) if and only if

∑_{j=1}^{∞} Q(B′φ_{j}^{p}, B′φ_{j}^{p}) < ∞

in which case

∥B∥_{L_{2}(H_{Q},Φ_{−p})}^{2} = ∑_{j=1}^{∞} Q(B′φ_{j}^{p}, B′φ_{j}^{p}).

Consider the Φ′− valued diffusion equation

dX_{t} = F (t, X_{t})dt + G(t, X_{t})dW_{t}, t ∈ R,  (2.5)

where F : R × Φ′ → Φ′, G : R × Φ′ → L(Φ′, Φ′) are measurable mappings.

We introduce the following assumptions on the coefficients (F, G) :

Assumption (A). There exists a p_{0} such that for every p ≥ p_{0} there are q ≥ p, θ = θ(p, q) > 0 and

β_{1} = β_{1}(p, q) < 0, β_{2} = β_{2}(p, q) > 0 , β_{1} + β_{2} < 0,

such that : (a1) (Continuity): For every t ∈ R,

F (t, .) : Φ_{−p} → Φ_{−q}, G(t, .) : Φ_{−p} → L_{2}(H_{Q}, Φ_{−p}),

are continuous. (a2) (Coercivity): For every t ∈ R, φ ∈ Φ

2F (t, φ) [θ_{p}φ] ≤ θ(1+ ∥ φ ∥_{−p}^{2}).

(a3) (Linear growth and boundedness): For every t ∈ R , φ ∈ Φ_{−p}

∥ F (t, φ) ∥_{−q}^{2}≤ θ(1+ ∥ φ ∥_{−p}^{2}),
∥ G(t, φ) ∥_{L_{2}(H_{Q},Φ_{−p})}^{2}≤ θ.

(a4) (Monotonicity): For every t ∈ R, φ_{1}, φ_{2} ∈ Φ_{−p}

< F (t, φ_{1}) − F (t, φ_{2}), φ_{1} − φ_{2} >_{−q}≤ β_{1} ∥ φ_{1} − φ_{2} ∥_{−q}^{2},
∥G(t, φ_{1}) − G(t, φ_{2})∥_{L_{2}(H_{Q},Φ_{−q})}^{2} ≤ β_{2} ∥ φ_{1} − φ_{2} ∥_{−p}^{2} .

We point out that for p_{1} ≥ p_{0} large enough we have for every p ≥ p_{1} that there exists \sqrt{Q_{p}} ∈ L(Φ_{p}), self-adjoint and nonnegative, with the following properties (see Lemma 3.2.2 of [12]):

(i) For every φ, ψ ∈ Φ,

Q(φ, ψ) = ⟨\sqrt{Q_{p}}φ, \sqrt{Q_{p}}ψ⟩_{p} .

(ii) H_{Q} is the image of \sqrt{Q_{p}}′ which is an isometry from Φ_{−p} to H_{Q}. Also, we choose p_{1} large enough such that p_{1} < q_{1} < p_{2}, where p_{1} < q_{1} is given by Assumption (A), q_{1} < p_{2} is such that the canonical injection i_{p_{2}q_{1}} : Φ_{p_{2}} → Φ_{q_{1}} is Hilbert-Schmidt.

Definition 2.4. A Φ′–valued F_{t}–progressively measurable process {X_{t}}_{t∈R} is a solution of (2.5) if for every s ≤ t we have

X_{t} = X_{s} + ∫_{s}^{t} F (u, X_{u})du + ∫_{s}^{t} G(u, X_{u})dW_{u}.  (2.6)

Definition 2.5. A solution {X_{t}}_{t∈R} of (2.5) is bounded in mean square (or L^{2}–bounded) if for all φ ∈ Φ

sup_{t∈R} E [X_{t}[φ]^{2}] < ∞,  (2.7)

or equivalently if there exists m ≥ 1 such that

sup_{t∈R} E[∥ X_{t} ∥_{−m}^{2}] < ∞.  (2.8)

Theorem 2.6 ([19]). Under the assumption (A) the equation (2.5) has a unique L^{2}–bounded continuous solution {X_{t}}_{t∈R}. Moreover X ∈ C(R, Φ_{−p_{2}}),

sup_{t∈R} E[∥ X_{t} ∥_{−p_{2}}^{2}] < ∞,  (2.9)

and if {X_{k}(t)}_{t≥−k} is the solution of the SDEs

X_{k}(t) = ∫_{−k}^{t} F (u, X_{k}(u))du + ∫_{−k}^{t} G(u, X_{k}(u))dW_{u}, t ≥ −k  (2.10)

then for every t ∈ R, X_{k}(t) ⟼_{k→∞}^{L^{2}} X_{t}.

3 Main Result

Before presenting the main result of this paper we make the following: Assumption (B): (b1) The assumption (A) is satisfied. (b2) (Lipschitz property): for every t ∈ R, φ_{1}, φ_{2} ∈ Φ_{−p}

∥F (t, φ_{1}) − F (t, φ_{2})∥_{−q}^{2} ≤ θ ∥ φ_{1} − φ_{2} ∥_{−p}^{2} .

Theorem 3.1. Assume (B) is satisfied. Let {X_{t}^{n}}_{t∈R}, n ≥ 1, be the unique bounded continuous solution with values in C (R, Φ_{−p_{2}}) of the Φ′-valued diffusion equation

dX_{t}^{n} = F (t, X_{t}^{n})dt + \frac{1}{\sqrt{n}}G(t, X_{t}^{n})dW_{t}, t ∈ R.  (3.11)

Then the sequence {X^{n}}_{n} satisfies the LDP on C (R, Φ_{−p_{2}}) with the good rate function

I(x) = inf \{\frac{1}{2} ∥h∥_{H}^{2} : h ∈ H, x = x^{h}\} ,  (3.12)

where, for h ∈ H, x^{h} is the unique bounded continuous solution of the deterministic integral equation

dx(t) = [F (t, x(t)) + G(t, x(t))ḣ(t)] dt, t ∈ R.  (3.13)

In order to prove this theorem we first present the following results. The idea is to prove large deviations for an appropriate approximating sequence so that the criterion of Theorem 2.1 can be applied.

Proposition 3.2. Assume (A) holds Then:

(a) For every h ∈ H the equation (3.13) has a unique bounded continuous solution x^{h}. Moreover x^{h} ∈ C (R, Φ_{−q_{1}}) and

sup_{t∈R} ∥x^{h}(t)∥_{−q_{1}}^{2} ≤ C (1 + ∥h∥_{H}^{2}) , C = C(θ(p_{1}, q_{1})).  (3.14)

(b) The application

h → x^{h} : H → C(R, Φ_{−q_{1}})

is continuous.

Proof. (a) Define

F^{h}(t, x) = F (t, x) + G(t, x)ḣ(t), t ∈ R, x ∈ Φ′.

By Theorem 10.2.7 in [12], it follows that for every a ∈ R, the deterministic integral equation

x_{a}^{h}(t) = ∫_{a}^{t} F^{h}(s, x_{a}^{h}(s))ds, t ≥ a,  (3.15)

has a unique solution in C ([a, ∞) , Φ_{−p_{1}}) . Let ε > 0 be such that β_{1} + ε < 0. Next from (a_{4}) we have that for x ∈ Φ_{−p_{1}},

2 ⟨F (t, x), x⟩_{−q_{1}} ≤ 2β_{1} ∥x∥_{−q_{1}}^{2} + 2 ⟨F (t, 0), x⟩_{−q_{1}}

≤ (2β_{1} + ε) ∥x∥_{−q_{1}}^{2} + ε^{−1} ∥F (t, 0∥_{−q_{1}}^{2} ≤ (2β_{1} + ε) ∥x∥_{−q_{1}}^{2} + θε^{−1},

2 ⟨G(t, x)ḣ(t), x⟩_{−q_{1}} ≤ ε ∥x∥_{−q_{1}}^{2} + ε^{−1} ∥G∥_{L^{2}(H_{Q},Φ_{−q_{1}})}^{2} ∥ḣ(t)∥_{H_{Q}}^{2} ,

and hence

2 ⟨F^{h}(t, x), x⟩_{−q_{1}} ≤ 2 (β_{1} + ε) ∥x∥_{−q_{1}}^{2} + C_{1} (1 + ∥ḣ(t)∥_{H_{Q}}^{2}) .  (3.16)

On the other hand, since

∥x_{a}^{h}(t)∥_{−q_{1}}^{2} = 2 ∫_{a}^{t} ⟨F^{h}(s, x_{a}^{h}(s)), x_{a}^{h}(s)⟩ ds,

we deduce by using (3.16)

\frac{d}{dt} ∥x_{a}^{h}(t)∥_{−q_{1}}^{2} ≤ 2 (β_{1} + ε) ∥x_{a}^{h}(t)∥_{−q_{1}}^{2} + C_{1} (1 + ∥ḣ(t)∥_{H_{Q}}^{2}) ,

so that

∥x_{a}^{h}(t)∥_{−q_{1}}^{2} ≤ C_{1} ∫_{a}^{t} e^{2(β_{1}+ε)(t−s)} (1 + ∥ḣ(s)∥_{H_{Q}}^{2}) ds,

and thus

sup_{t≥a} ∥x_{a}^{h}(t)∥_{−q_{1}}^{2} ≤ −\frac{C_{1}}{2 (β_{1} + ε)} + C_{1} ∥h∥_{H}^{2} .  (3.17)

Then, since for all x, y ∈ Φ_{−p_{1}},

2 ⟨F^{h}(t, x) − F^{h}(t, y), x − y⟩_{−q_{1}} ≤ (β_{1} + ε + ε^{−1}β_{2} ∥ḣ(t)∥_{H_{Q}}^{2}) ∥x − y∥_{−q_{1}}^{2} and for a < b < t we have

x_{a}^{h}(t) − x_{b}^{h}(t) = x_{a}^{h}(b) + ∫_{b}^{t} [F^{h}(s, x_{a}^{h}(s)) − F^{h}(s, x_{b}^{h}(s))] ds. Then it follows by Itô’s formula, (3.17) and Gronwall’s lemma that

∥x_{a}^{h}(t) − x_{b}^{h}(t)∥_{−q_{1}}^{2} ≤ e^{(β_{1}+ε)(t−b)} ∥x_{a}^{h}(b)∥_{−q1}^{2} +

ε^{−1}β_{2} ∫_{b}^{t} e^{(β_{1}+ε)(t−s)} ∥ḣ(s)∥_{H_{Q}}^{2} ∥x_{a}^{h}(s) − x_{b}^{h}(s)∥_{−q_{1}}^{2} ds,

\overrightarrow{lim_{a,b→−∞}} ∥x_{a}^{h}(t) − x_{b}^{h}(t)∥_{−q_{1}}^{2} ≤

ε^{−1}β_{2} ∫_{−∞}^{t} e^{(β_{1}+ε)(t−s)} ∥ḣ(s)∥_{H_{Q}}^{2} \overrightarrow{lim_{a,b→−∞}} ∥x_{a}^{h}(s) − x_{b}^{h}(s)∥_{−q_{1}}^{2} ds ≤

ε^{−1}β_{2} ∫_{−∞}^{t} ∥ḣ(s)∥_{H_{Q}}^{2} \overrightarrow{lim_{a,b→−∞}} ∥x_{a}^{h}(s) − x_{b}^{h}(s)∥_{−q_{1}}^{2} ds.

Therefore

\overrightarrow{lim_{a,b→−∞}} ∥x_{a}^{h}(t) − x_{b}^{h}(t)∥_{−q_{1}}^{2} = 0,

and

lim_{a→−∞} x_{a}^{h}(t) = x^{h}(t) in Φ_{−q_{1}} for all t ∈ R.

Since for t_{2} > t_{1} > a it holds that

x_{a}^{h}(t_{2}) − x_{a}^{h}(t_{1}) = ∫_{t_{1}}^{t_{2}} F^{h}(s, x_{a}^{h}(s))ds,

by letting a → −∞ we get

x^{h}(t_{1}) − x^{h}(t_{2}) = ∫_{t_{1}}^{t_{2}} F^{h}(s, x^{h}(s))ds,  (3.18)

i.e., {x^{h}(t)}_{t∈R} is a solution of (3.13). Furthermore (3.18) shows that x^{h} is continuous and using (3.17) we obtain (3.14). Uniqueness. Let x, y be bounded and continuous solutions of (3.13). Without loss of generality we can assume that x, y take values in Φ_{−p_{1}}. Then, proceeding as above we obtain

∥x(t) − y(t)∥_{−q_{1}}^{2} ≤ ε^{−1}β_{2} ∫_{−∞}^{t} ∥ḣ(s)∥_{H_{Q}}^{2} ∥x(s) − y(s)∥_{−q_{1}}^{2} ds,

and consequently x(t) = y(t) for all t ∈ R.

(b) If a < b and h_{n} → h in H, then

sup_{a≤t≤b} ∥x^{h_{n}}(t) − x^{h}(t)∥_{−q_{1}}^{2} ≤

ε^{−1}β_{2} ∫_{−∞}^{b} [∥ḣ(s)∥_{H_{Q}}^{2} sup_{a≤u≤s} ∥x^{h_{n}}(u) − x^{h}(u)∥_{−q_{1}}^{2} + θ ∥ḣ_{n}(s) − ḣ(s)∥_{H_{Q}}^{2}] ds,

and therefor

lim_{n→∞} sup_{a≤t≤b} ∥x^{h_{n}}(t) − x^{h}(t)∥_{−q_{1}}^{2} = 0. ■

Lemma 3.3. Suppose (A) is satisfied. Then for every r > 0 the family F_{r} = {x_{a}^{h}(.)}_{∥h∥_{H}≤r,a∈R} is relatively compact in C(R, Φ_{−p_{2}}) (by definition

x_{a}^{h}(t) = 0 if t ≤ a). In particular if h_{k} → h in H, then

x_{−k}^{h_{k}} → x^{h} in C(R, Φ_{−p_{2}}).

Proof. From (3.17) it follows that {x_{a}^{h}(t)}_{∥h∥_{T} ≤r,a,t∈R} is relatively compact in Φ_{−p_{2}}. Next if a ≤ s < t then it follows that

sup_{∥h∥_{T} ≤r} ∥x_{a}^{h}(t) − x_{a}^{h}(s)∥_{−p_{2}}^{2} ≤ C(t − s),

where C is independent of a ∈ R. So by Arzela-Ascoli F_{r} is relatively compact in C(R, Φ_{−p_{2}}) . Now if h_{k} → h in H, h_{k}, h ∈ H , then there exists a subsequence {h_{k(p)}} such that y_{p}(.) = x_{−k(p)}^{^{h}k(p)}(.) → y(.) in C(R, Φ_{−p_{2}}). For s < t, by passing to the limit in the equality

y_{p}(t) − y_{p}(s) = ∫_{s}^{t} [F (u, y_{p}(u)) + G(u, y_{p}(u))ḣ(u)] du,

we obtain that y is a solution to (3.13) and by uniqueness this yields y = x^{h}. ■

For ξ ∈ Φ′, h ∈ H^{T} we consider the skeleton

x(t) = ξ + ∫_{0}^{t} [F (u, x(u)) + G(u, x(u))ḣ(u)] du, t ∈ [0, T] .  (3.19)

Remark 3.4 ([12]-Theorem 10.2.1). The triplet (i, H^{T} , C ([0, T] , Φ_{−q_{1}})), where i : H^{T} → C ([0, T] , Φ_{−q_{1}}) is the canonical injection, is an abstract Wiener space. In particular, for every r > 0, the set K_{T,r} = {h ∈ H^{T} : ∥h∥_{H^{T}} ≤ r} is compact in C ([0, T] , Φ_{−q_{1}}) (see [8], [11] or [13]).

Remark 3.5 ([12]-Theorem 10.2.7). Let ξ ∈ Φ_{−_{q_{1}}}. Under (A) the equation (3.19) has a unique bounded solution {x_{T,ξ}^{h}(t)}_{t∈[0,T]} in C ([0, T] , Φ_{−p_{2}}) and the application

h → x_{T,ξ}^{h}(.) : H^{T} → C ([0, T] , Φ_{−p_{2}}) ,

is continuous. Let I_{T,ξ} : C ([0, T] , Φ_{−p_{2}}) → R̄_{+}, be defined by

I_{T,ξ}(x) = inf \{\frac{1}{2} ∥h∥_{H^{T}}^{2} : h ∈ H^{T} , x = x_{T,ξ}^{h}\} .  (3.20)

and consider the sequence of diffusion equations in Φ′

X_{t}^{n} = ξ + ∫_{0}^{t} F (s, X_{s}^{n})ds + \frac{1}{\sqrt{n}} ∫_{0}^{t} G(s, X_{s}^{n})dW_{s}, t ∈ [0, T] .  (3.21)

Theorem 3.6 ([12], [21]). Under (B) the sequence X^{n} satisfies the LDP in C ([0, T] , Φ_{−p_{2}}) with the good rate function I_{T,ξ} given by (3.20).

Lemma 3.7. The application

(ξ, x) → I_{T,ξ}(x) : Φ_{−p_{2}} × C ([0, T] , Φ_{−p_{2}}) → R̄_{+},

is lower semicontinuous. Proof. Let (ξ_{n}, x_{n}) → (ξ, x) and prove that

I_{T,ξ}(x) ≤ \underrightarrow{lim}_{n→∞} I_{T,ξ_{n}}(x_{n}) = α.

If α = 0 there is nothing to prove. Assume α > 0 and let δ > 0, then there exists a subsequence n (k) such that

I_{T,ξ_{n(k)}}(x_{n(k)}) < α + δ, ∀k.

Choose h_{n(k)} ∈ H^{T} such that

\frac{1}{2} ∥h_{n(k)}∥_{H^{T}}^{2} ≤ I_{T,ξ_{n(k)}}(x_{n(k)}) + δ ≤ α + 2δ, ∀k.  (3.22)

Without loss of generality we can assume that

ḣ_{n(k)} → ḣ weakly in L^{2} ([0, T] , H_{Q}) .

Also, from Remark 3.4 we can suppose that h_{n(k)}(.) → h(.) in C ([0, T] , Φ_{−p_{2}}) . Then it follows that x = x_{T,ξ}^{h} and by (3.22) we have

\frac{1}{2} ∥h∥_{H^{T}}^{2} ≤ α, i.e. I_{T,ξ}(x) ≤ α. ■

The next lemma follows along the same lines as Proposition 2.5 of [20].

Lemma 3.8. Suppose that F = {F_{t}}_{t≥a} , G = {G_{t}}_{t≥a} are Φ′ -valued, L (Φ′, Φ′) -valued predictable processes and that the Φ′ -valued process {Z_{t}}_{t≥a} = {Z_{t}^{n,F,G}}_{t≥a} , n ≥ 1, satisfies the equality

dZ_{t} = F_{t}dt + \frac{1}{\sqrt{n}}G_{t}dW_{t}, ∀t ≥ a.

Moreover, suppose that there exists a p_{0} such that for every p ≥ p_{0} there exists q ≥ p, with:

(1) F, G and Z take values in Φ_{−q}, L_{2}(H_{Q}, Φ_{−p}) and Φ_{−q} respectively.

(2) E [∥Z_{a}∥_{−q}^{2n}] < ∞.

(3) For some constants α, β ∈ R, M > 0, we have P -a.s. for all t ≥ 0,

⟨Z_{t}, F_{t}⟩_{−q} ≤ α + β ∥Z_{t}∥_{−q}^{2} , ∥G_{t}∥_{L_{2}(H_{Q},Φ_{−p})}^{2} ≤ M.  (3.23)

For δ > 0 define τ = inf {t ≥ a : ∥Z_{t}∥_{−q} ≥ δ} . Then

E [∥Z_{t∧τ}∥_{−q}^{2n}] ≤ E [∥Z_{a}∥_{−q}^{2n}] e^{2βn(t−a)} +  (3.24)
2n (α + M^{2}) ∫_{a}^{t} e^{2βn(t−s)}E [∥Z_{s∧τ}∥_{−q}^{2(n−1)}] ds, ∀t ≥ a.

In particular if Z_{a} = 0, α ≥ 0, β < 0, then

sup_{t≥a} E [∥Z_{t}∥_{−q}^{2n}] ≤ C (−\frac{α + M^{2}}{β} )^{n} ,  (3.25)

for some constant C > 0 independent of a and n. Moreover if α = 0 and

∥G_{t}∥_{−p}^{2} ≤ γ ∥Z_{t}∥_{−p}^{2} , ∀t ≥ a,  (3.26)

for some constant γ > 0, then

E [∥Z_{t∧τ}∥_{−q}^{2n}] ≤ E [∥Z_{a}∥_{−q}^{2n}] e^{2n(β+γ)(t−a)}, ∀t ≥ a.  (3.27)

Proof of Theorem 3.1. First note that by Proposition 3.2 the equation (3.13) has a unique bounded continuous solution x^{h} ∈ C (R, Φ_{−p_{1}}) . From the Dawson-Gärtner LDP for projective limits (see [7]-Theorem 4.6.1) it is enough to prove that for every a, b ∈ R, the sequence {X_{t}^{n}}_{t∈[a,b]} satisfies the LDP in C ([a, b] , Φ_{−p_{2}}) with the good rate function

I_{a,b}(x) = inf \{\frac{1}{2} ∥h∥_{H}^{2} : h ∈ H, x(t) = x^{h}(t), ∀t ∈ [a, b]\} .  (3.28)

Now we use Theorem 2.1 in order to prove that the sequence of Φ_{−p_{2}} -valued random variables {X_{a}^{n}}_{n} satisfies the LDP with the rate function

I_{a}(ξ) = inf \{\frac{1}{2} ∥h∥_{H}^{2} : h ∈ H, ξ = x^{h}(a)\} .  (3.29)

Note that by Remark 3.4 the function Λ(h) = \frac{1}{2} ∥h∥_{H}^{2} , h ∈ H, Λ(h) = ∞, h ∉ H, is a good rate function. From Lemma 3.3 it follows that assumption (1) of Theorem 2.1 is satisfied. Next, let X_{k}^{n} be given by

X_{k}^{n}(t) = ∫_{−k}^{t} F (s, X_{k}^{n}(s))ds + \frac{1}{\sqrt{n}} ∫_{−k}^{t} G(s, X_{k}^{n}(s))dW_{s}, t ≥ −k.

Using Lemma 3.8 it follows as in [20] that X_{k}^{n} is an exponentially good approximation of X^{n}. By definition we have that

X_{t}^{n} = X_{a}^{n} + ∫_{a}^{t} F (s, X_{s}^{n})ds + \frac{1}{\sqrt{n}} ∫_{a}^{t} G(s, X_{s}^{n})dW_{s}, t ∈ [a, b] .

From Theorem 3.6, Lemma 3.7, (3.29) and Theorem 2.2 we deduce that the pair (X_{a}^{n}, {X_{t}^{n}}_{t∈[a,b]}) satisfies the LDP on Φ_{−p_{2}}× C ([a, b] , Φ_{−p_{2}}) with the good rate function

I(ξ, f) = I_{a}(ξ) + I_{a,b,ξ}(f),

where

I_{a,b,ξ}(f) = inf \{\frac{1}{2} ∥h∥_{H^{a,b}}^{2} : h ∈ H^{a,b}, f = x_{a,b,ξ}^{h}\} ,

(here H^{a,b} is defined as H^{T} with [0, T] replaced by [a, b]). The result then follows by the standard contraction principle. ■

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