SELF-INTERSECTION LOCAL TIME OF THE FLUCTUATION LIMIT OF A MULTITYPE BRANCHING PARTICLE SYSTEM

Ekaterina Todorova Centro de Investigación en Matemáticas

Abstract

We give a necessary and sufficient condition for existence and continuity of selfintersection local time (SILT) of the fluctuation limit of a multitype branching particle system in R^{d}, with particles evolving according to symmetric stable motions with different stability parameters. We show that existence of SILT is determined by the “most mobile” of the stable motions.

1 Introduction

Let X = (X(t))_{t∈[0,1]} be a centered, continuous, Gaussian process with values in the space S′(R^{d}) of tempered distributions on R^{d}. An intuitive definition of self-intersection local time (SILT) of X up to time t ∈ [0, 1] is given by the formal expression

∫_{0}^{t} ∫_{0}^{t}⟨X(s) ⊗ X(r), δ(x − y)φ(x)⟩ ds dr,  (1.1)

where ⊗ denotes the tensor product in S′(R^{d}), φ ∈ S(R^{d}) (the C^{∞} rapidly decreasing functions), δ is the Dirac distribution, and ⟨·, ·⟩ stands for duality. Since δ(x − y)φ(x) ∉ S(R^{2d}), the first question is how to give a rigorous meaning to (1.1). This problem was studied by Adler, Feldman and Lewin (1991), and Adler and Rosen (1993), when X is the Brownian and the α-stable density process, respectively. In Adler et al. (1991) and Adler and Rosen (1993) it is shown that the SILT of X exists if and only if d < 2α, and for α = 2 the SILT process, when it exists, has cadlag paths.

Motivated by their work, Bojdecki and Gorostiza (1995) gave a rigorous definition of (1.1) for a wide class of S′(R^{d})-valued Gaussian processes and a criterion for existence and continuity of SILT in terms of the covariance of a given process. They applied this criterion to various examples of generalized processes, and in particular, they proved that the SILT of the α-stable density process exists and is continuous if and only if d < 2α. The methods of proof in Adler, Feldman and Lewin (1991), Adler and Rosen (1993), and Bojdecki and Gorostiza (1995,1996,1997), make use in a fundamental way of the Chapman-Kolmogorov formula and the self-similarity of α-stable processes.

Todorova (1997), and Gorostiza and Todorova (1998) obtained an existence and continuity result for SILT of a more general density process that corresponds to a system of particles of two types, where the partices of type i, i = 1, 2, move according to a symmetric α_{i}-stable process, and each particle switches back and forth between the two types with respective exponential waiting times with parameters V_{i}. It is shown that the SILT exists and is continuous if and only if d < 2 min{α_{1}, α_{2}}, thus generalizing the above results. The intuitive interpretation of this result is that the existence of SILT is determined by the “most mobile” of the stable motions. The main difficulty in determining the existence and continuity of SILT comes from the presence of different types of particle motions, which does not allow a direct application of the criterion in Bojdecki and Gorostiza (1995) because this density process is not Markovian nor self-similar.

In this paper we consider the SILT of a high density fluctuation limit of a system of particles of k types, k ≥ 2, in which a particle of type i follows a symmetric α_{i}-stable motion for an exponential lifetime of parameter V_{i}, after which it branches according to a finite variance, multitype branching law. Due to the presence of a genuine multitype branching (not only of type mutation, as in the above work), the covariance functional of the process is more complicated and the implementation of the criteria quoted above is more difficult.

Under usual conditions for the Markov chain of types, we show that the SILT exists and is continuous if and only if d < 2 min{α_{1}, ..., α_{k}}; that is, again the “ most mobile” motion determines existence of SILT. Clearly, in the case k = 2 and type mutation, this result reduces to the previous one.

The organization of this paper is as follows. In Section 2 we recall the definition of SILT of generalized Gaussian processes given by Bojdecki and Gorostiza (1995), as well as their criterion for existence and continuity of SILT. In Section 3 we introduce the particle system which we consider here, and state our main result on the existence of SILT (Theorem 3.2) which is proved in Section 4.

2 A general criterion on existence and continuity of SILT for S′(R^{d})-valued processes

We will condense the main result from Bojdecki and Gorostiza (1995). We refer the reader to that paper for additional information and details.

Let F denote the class of non-negative symmetric C^{∞} functions f on R^{d} with bounded support, and such that f(0) > 0 and ∫ f(x) dx = 1. For f ∈ F, ε > 0 and φ ∈ S(R^{d}) we define

f_{ε}(x) = ε^{−d}f (\frac{x}{ε}) , x ∈ R^{d},

and

Φ_{ε,φ}^{f}(x, y) = f_{ε}(x − y)φ(x), x, y ∈ R^{d}.

Note that Φ_{ε,φ}^{f} ∈ S(R^{2d}) and Φ_{ε,φ}^{f} approximates δ(x − y)φ(x) as ε → 0. In order to give a meaning to (1.1), the idea is to replace δ(x − y)φ(x) by Φ_{ε,φ}^{f}, so that it makes sense, and take the limit as ε → 0. But this is not enough; for existence of a limit it is also necessary to replace X(s) ⊗ X(r) by the Wick product : X(s) ⊗ X(r) :, which is an S′(R^{2d})-valued random field such that

⟨: X(s) ⊗ X(r) :, φ ⊗ ψ⟩ = ⟨X(s), φ⟩⟨X(r), ψ⟩ − E(⟨X(s), φ⟩⟨X(r), ψ⟩).

This leads to defining an approximate SILT L_{ε}^{f}(t) by

⟨L_{ε}^{f}(t), φ⟩ = ∫_{0}^{t} ∫_{0}^{t}⟨: X(s) ⊗ X(r) :, Φ_{ε,φ}^{f}⟩ ds dr, t ∈ [0, 1], φ ∈ S(R^{d}),

which is a continuous S′(R^{d})-process.

We can now give a precise meaning to (1.1).

Definition 2.1. For a given continuous centered Gaussian S′(R^{d})-valued process X = (X(t))_{t∈[0,1]}, if there exists an S′(R^{d})-valued process L = (L(t))_{t∈[0,1]} such that for any t ∈ [0, 1], φ ∈ S(R^{d}) and f ∈ F,

⟨L_{ε}^{f}(t), φ⟩ → ⟨L(t), φ⟩

in L^{2} as ε → 0, then L is called the self-intersection local time (SILT) of X.

Let K denote the covariance functional of X :

K(s, φ; t, ψ) = E (⟨X(s), φ⟩ ⟨X(t), ψ⟩) , φ, ψ ∈ S(R^{d}), s, t ∈ [0, 1].

For test functions Φ^{(1)}, Φ^{(2)} ∈ S(R^{d}) ⊗ S(R^{d}) of the form

Φ^{(1)} = ∑_{i=1}^{n} φ_{i}^{(1)} ⊗ ψ_{i}^{(1)}, Φ^{(2)} = ∑_{j=1}^{m} φ_{j}^{(2)} ⊗ ψ_{j}^{(2)}, φ_{i}^{(1)}, ψ_{i}^{(1)}, φ_{j}^{(2)}, ψ_{j}^{(2)} ∈ S(R^{d}),  (2.1)

and s, r, u, v ∈ [0, 1], we consider the functional

J_{s,r,u,v}(Φ^{(1)}, Φ^{(2)})  (2.2)
= ∑_{i=1}^{n} ∑_{j=1}^{m} (K(s, φ_{i}^{(1)}; u, φ_{j}^{(2)})K(r, ψ_{i}^{(1)}; v, ψ_{j}^{(2)}) + K(s, φ_{i}^{(1)}; v, ψ_{j}^{2})K(r, ψ_{i}^{(1)}; u, φ_{j}^{(2)})) ,

which is the covariance functional of the random field : X(s) ⊗ X(r) : for test functions of the given form.

We now state the existence and continuity criterion for SILT:

Theorem 2.2. 1. Given a continuous centered Gaussian S′(R^{d})-process X, assume that J_{s,r,u,v}(Φ^{(1)}, Φ^{(2)}) has a well defined extension on S(R^{2d}) × S(R^{2d}) such that

(i) The functional

(Φ^{(1)}, Φ^{(2)}) ↦ ∫_{[0,t]^{4}} J_{s,r,u,v}(Φ^{(1)}, Φ^{(2)}) ds dr du dv

is continuous on S(R^{2d}) × S(R^{2d}) for each t ∈ [0, 1].

(ii) J_{s,r,u,v}(Φ_{ε,φ}^{f}, Φ_{δ,φ}^{g}) converges to a finite limit as ε, δ → 0, for each f, g ∈ F, φ ∈ S(R^{d}), s, r, u, v ∈ [0, 1], and this limit does not depend on f, g.

(iii)

|J_{s,r,u,v}(Φ_{ε,φ}^{f}, Φ_{δ,φ}^{g})| ≤ G_{φ}(s, r, u, v)

for some measurable function G_{φ} on [0, 1]^{4} which depends on φ but is independent of ε, δ, f, g, and such that

∫_{[0,1]^{4}} G_{φ}(s, r, u, v) ds dr du dv < ∞

for each φ ∈ S(R^{d}).

Then the SILT L of the process X exists.

Assume in addition that

(iv) There exists a non-decreasing continuous function F on [0, 1] and a number γ > 0 such that for all t_{1}, t_{2} ∈ [0, 1], t_{1} < t_{2}, φ ∈ S(R^{d}),

∫_{[0,1]^{4}} (1_{[0,t_{2}]^{2}}(s, r) − 1_{[0,t_{1}]^{2}}(s, r)) (1_{[0,t_{2}]^{2}}(u, v) − 1_{[0,t_{1}]^{2}}(u, v)) G_{φ}(s, r, u, v) ds dr du dv

≤ C(φ) (F (t_{2}) − F (t_{1}))^{1+γ} ,

where C(φ) is a positive constant depending only on φ.

Then the SILT L is a continuous S′(R^{d})-process, and moreover L_{ε}^{f} converges weakly to L in C([0, 1], S′(R^{d})) as ε → 0.

2. Suppose that J_{s,r,u,v} satisfies condition (i) but

lim_{ε→0} ∫_{[0,1]^{4}} J_{s,r,u,v}(Φ_{ε,φ}^{f}, Φ_{ε,φ}^{f}) ds dr du dv = ∞

for some φ ∈ S(R^{d}) and f ∈ F. Then X does not have SILT.

3 The particle system and the result on existence of SILT for the fluctuation limit process

We consider a particle system of k types in R^{d}, k ≥ 2, that correspond to a symmetric α_{i}- stable motions, α_{i} ∈ (0, 2], i = 1, 2, ..., k. At time t = 0 the particles of type i are distributed according to a Poisson random field with intensity measure Kγ_{i}Λ, where K > 0, λ_{i} ≥ 0, and Λ denotes the Lebesgue measure on R^{d}. Each particle of type i moves according a symmetric α_{i}-stable process with exponential lifetime with parameter V_{i} > 0, and then branches into particles of k types, following a branching law p_{i} (p_{i}(j_{1}, ..., j_{k}) is the probability that the particle of type i produces j_{s} particles of type s, s = 1, ..., k). The offspring particles appear at the same place of branching of their progenitor, and move and branch following the same rules.

We define the first branching moments

m_{i}^{(1)}(h) = ∑_{j_{1},...,j_{k}≥0} j_{h}p_{i}(j_{1}, … j_{k}), i, h = 1, ..., k

and denote by M^{(1)} the matrix

M^{(1)} = (m_{i}^{(1)}(h))_{i,h=1,2,...,k}.

We define the second factorial moments

m_{i}^{(2)}(h, l) = ∑_{j_{1},...,j_{k}≥0} j_{h}(j_{l} − δ_{hl})p_{i}(j_{1}, …, j_{k}), i, h, l = 1, ..., k,

where δ_{hl}is the Kronecker δ, the functions

M_{i}^{(2)}(x, y) = ∑_{h,l=1}^{k}m_{i}^{(2)}(h, l)x_{h}y_{l}, x = (x_{1}, …, x_{k}), y = (y_{1}, …, y_{k}) ∈ R^{k}, i = 1, …, k, (3.1)

and the vector

M^{(2)} = (M_{1}^{(2)}, …, M_{k}^{(2)})^{t}

(A^{t} means the transpose matrix of A).

We denote

V = diag(V_{1}, …, V_{k})

and

Δ_{α} = diag(Δ_{α_{1}}, … Δ_{α_{k}}),

where Δ_{α_{i}} is the generator of the symmetric α_{i}-stable motion, i = 1, ..., k.

We define the semigroup U(t), t ≥ 0, in (L^{1}(R^{d}))^{k} with generator U = Δ_{α}+V (M^{(1)}−I). We assume the following conditions on the branching law: (A) The mean matrix M^{(1)} es irreducible and apperiodic. (B) The second factorial moments are finite. We denote by N_{i}^{K}(t, A) the number of particles of type i which belong to the set

A ∈ B(R^{d}) at time t, i = 1, ..., k ( N_{i}^{K}(t) is a random point measure), and by M_{i}^{K} the corresponding fluctuation process: M^{K} = (M_{1}^{K}, ..., M_{k}^{K}), where

M_{i}^{K}(t) = K^{−1/2}(N_{i}^{K}(t) − EN_{i}^{K}(t)), t ∈ [0, 1], i = 1, ..., k.

We will use the notation

Φ ⊙ Ψ = (φ_{1}ψ_{1}, ..., φ_{k}ψ_{k}),

where Φ = (φ_{1}, ..., φ_{k}), Ψ = (ψ_{1}, ..., ψ_{k}), ϕ_{i}, ψ_{i} ∈ S(R^{d}), and

⟨Γ, Φ⟩ = ∑_{i=1}^{k} γ_{i} ∫_{R^{d}} φ_{i}(x)dx

where Γ = (γ_{1}Λ, ..., γ_{k}Λ).

The following result is known (López-Mimbela, 1992):

Theorem 3.1. When K → ∞, M^{K} converges weakly to the process M = (M_{1}, ..., M_{k}) in the Skorohod space D([0, 1], (S′(R^{d}))^{k}), where M is a continuous centered Gaussian Markov process with covariance functional K(s, Φ; t, Ψ) = E(⟨M(s), Φ⟩⟨M(t), Ψ⟩) given by

K(s, Φ; t, Ψ) = ⟨Γ, U(s)(Φ ⊙ U(t − s)Ψ)⟩  (3.2)
+ ⟨Γ, ∫_{0}^{s} U(v)V M^{(2)}(U(s − v)Φ, U(t − v)Ψ) dv, ⟩
s, t ∈ [0, 1], s ≤ t, Φ, Ψ ∈ (S(R^{d}))^{k}.

The process M is the high-density fluctuation limit of the given branching particle system.

Remark. The first term in (3.2) is due to the type mutation, while the second term is due to the genuine branching. This is the reason for the more complicated expression of the covariance functional here than in the model of type mutation in Gorostiza and Todorova (1998).

Our goal is to find conditions for the existence and continuity of the SILT of the process

X = M_{1} + ... + M_{k},

which allows the study the influence of the branching in the particle system. Note that M_{1}, ..., M_{k} are Markovian processes, but X is not Markovian (from Lemma 4.1 we see that the covariance functional of X does not satisfy the condition Cov(⟨X_{r}, φ⟩, ⟨X_{t}, ψ⟩) = Cov(⟨X_{r}, φ⟩, ⟨X_{s,t}, ψ⟩) for some random variable X_{s,t} in S′(R^{d}), r ≤ s < t, which is the condition that caracterizes the Markov property of Gaussian processes in S′(R^{d}) (e.g. Fernández, 1991).

Our main result is the following theorem.

Theorem 3.2. We assume conditions (A) and (B). Then

a) If d < 2 min{α_{1}, ..., α_{k}}, the SILT of X exists and is a continuous process.

b) If d ≥ 2 min{α_{1}, ..., α_{k}}, the SILT of X does not exist.

Remarks. 1. When α_{1} = α_{2} = ... = α_{k}, this result coincides with those of Adler, Feldman and Lewin (1991), Adler and Rosen (1993) (without branching), and Bojdecki and Gorostiza (1995,1996) (with branching).

2. We can prove analogously that the SILT of the process M_{i} exists if and only if d < 2α_{i}, i = 1, ..., k, and it is a continuous process.

4 Proof of Theorem 3.1.

For simplicity in the expressions we assume k = 2. In the general case the proof is analogous.

We will prove that the conditions of Theorem 2.2 are satisfied using the methods of Gorostiza and Todorova (1998). First we prove the theorem for the case of stationary distribution of the Markov chain of types, and then we consider the general case by comparing with the stationary case. We introduce first some definitions.

Let denote by B ≡ (ξ(t), τ(t))_{t≥0} the basic process, or the “expectation process,” of our branching particle system. B is a Markov process in R^{d} × {1, 2} where the first component corresponds to the position and the second component corresponds to the type of the particle at time t. The type component (τ_{t}) is a Markov process in {1, 2} with generador A = V (M^{(1)} −I), A = (A_{ij})_{i,j=1,2}, and the position component (ξ_{t}) follows a symmetric α_{i}-stable motion while τ_{t} = i.

For U(t) = (U_{ij}(t))_{i,j=1,2} acting on funtions of the form F (x, i) = φ(x)1_{{j}}(i), φ ∈ S(R^{d}), x ∈ R^{d}, we write

U_{ij}(t)φ(x) = U(t)F (x, i).  (4.1)

Thus

U_{ij}(t)φ(x) = ∫ φ(y)h_{t}^{(ij)}(x, y) dy,  (4.2)

where

h_{t}^{(ij)}(x, y) dy = P [ξ(t) ∈ dy |ξ(0) = x, τ(0) = i, τ(t) = j] P [τ(t) = j |τ(0) = i] . (4.3)

In particular the operators U_{ij}(t), i, j = 1, 2, t ≥ 0, are positive. For all t > 0 the functions h_{t}^{(ij)}(x, ·) are continuous and integrable, and satisfy

h_{t}^{(ij)}(x, y) = h_{t}^{(ij)}(x − y) and ∫ h_{t}^{(ij)}(x) dx ≤ 1.  (4.4)

We define the operators

U_{i}(t) = γ_{1}U_{1i}(t) + γ_{2}U_{2i}(t), i = 1, 2.  (4.5)

Hence

U_{i}(t)φ(x) = ∫ φ(y)h_{t}^{(i)}(x, y) dy, i = 1, 2.  (4.6)

where

h_{t}^{(i)}(x, y) = γ_{1}h_{t}^{(1i)}(x, y) + γ_{2}h_{t}^{(2i)}(x, y), i = 1, 2.  (4.7)

For t > 0 the funtions h_{t}^{(i)}(x, ·) are continuous and integrable.

We introduce the functions p_{t}^{(i)}(x, y), i = 1, 2 :

p_{t}^{(i)}(x, y) = h_{t}^{(i1)}(x, y) + h_{t}^{(i2)}(x, y);  (4.8)

hence

(U_{i1}(t) + U_{i2}(t))φ(x) = ∫ φ(y)p_{t}^{(i)}(x, y) dy.  (4.9)

From the above we have that for all t > 0 the functions p_{t}^{(i)}(x, ·) are continuous and integrable.

From (4.3) we have that

∫ φ(y)p_{t}^{(i)}(x, y) dy = E [φ(ξ(t))|ξ(0) = x, τ(0) = i] ,

therefore, by a slight abuse of language we refer to p_{t}^{(i)}(x, y) as the transition densities of the position component of the basic process (altough it is not a Markov process).

Lemma 4.1. The covariance functional of the process X is given by

K(s, φ; t, ψ) = ∑_{i,j=1}^{2} ∫ U_{i}(s) [φ(x) ∫ p_{t−s}^{(i)}(x, y)ψ(y) dy] (x) dx

+ ∑_{j,l=1}^{2} ∫_{0}^{s} ∫ L_{jl}(s) [φ(y) ∫ p_{s−v}^{(j)}(x, y) dy ∫ ψ(w)p_{t−v}^{(l)}(x, w) dw] (x) dx dv,  (4.10)

where

U_{i}(s) = γ_{1}U_{1i}(s) + γ_{2}U_{2i}(s)

and

L_{jl}(s) = ∑_{i=1}^{2} γ_{i} (V_{1}m_{1}^{(2)}(j, l)U_{i1}(s) + V_{2}m_{2}^{(2)}(j, l)U_{i2}(s)) , j, l = 1, 2.

Proof. We apply formula (3.2) with Φ = (φ, φ) and Ψ = (ψ, ψ), using that

K(s, φ; t, ψ) = E(⟨M_{1}(s) + M_{2}(s), φ⟩⟨M_{1}(t) + M_{2}(t), ψ⟩)
= E(⟨(M_{1}(s), M_{2}(s), (φ, φ)⟩⟨(M_{1}(t), M_{2}(t), (ψ, ψ)⟩).

From Lemma 5.1 in Gorostiza and Todorova (1998), we obtain the expression for the first term of (3.2), which is the first term of (4.10).

We will obtain the expression for the second term of (3.2). First we compute the expression for M^{(2)}(U(s − v)Φ, U(t − v)Ψ)). Using (4.9) we obtain

M^{(2)}(U(s − v)Φ, U(t − v)Ψ)) = [{M_{1}^{(2)} (U(s − v)Φ, U(t − v)Ψ) \atop M_{2}^{(2)} (U(s − v)Φ, U(t − v)Ψ)}]

= [{M_{1}^{(2)} (\left[{(U_{11} + U_{12})(s − v)φ \atop (U_{21} + U_{22})(s − v)φ}\right] , \left[{(U_{11} + U_{12})(t − v)ψ \atop (U_{21} + U_{22})(t − v)ψ}\right]) \atop M_{2}^{(2)} (\left[{(U_{11} + U_{12})(s − v)φ \atop (U_{21} + U_{22})(s − v)φ}\right] , \left[{(U_{11} + U_{12})(t − v)ψ \atop (U_{21} + U_{22})(t − v)ψ}\right])}]

= [ \begin{matrix}A_{1} \\ A_{2} \end{matrix} ] ,

where

A_{1} = ∑_{i,j=1}^{2} m_{1}^{(2)}(i, j)[(U_{i1} + U_{i2})(s − v)φ][(U_{j1} + U_{j2})(t − v)ψ],

A_{2} = ∑_{i,j=1}^{2} m_{2}^{(2)}(i, j)[(U_{i1} + U_{i2})(s − v)φ][(U_{j1} + U_{j2})(t − v)ψ].

Hence,

U(v)V M^{(2)}(U(s − v)Φ, U(t − v)Ψ) = [ \begin{matrix}V_{1}U_{11}(v)A_{1} + V_{2}U_{12}(v)A_{2} \\ V_{1}U_{21}(v)A_{1} + V_{2}U_{22}(v)A_{2} \end{matrix} ] .

We substitute the above expression in the second term of (3.2). From (4.9) we have for i, j = 1, 2,

(U_{i1} + U_{i2})(s − v)φ(x) = ∫ φ(y)p_{s−v}^{(i)}(x, y) dy,
(U_{j1} + U_{j2})(t − v)ψ(x) = ∫ ψ(w)p_{t−v}^{(j)}(x, w) dw, i, j = 1, 2.

Using this and grouping the terms which contain ∫ φ((y)p_{s−v}^{(i)}(x, y) dy ∫ ψ(w)p_{t−v}^{(j)}(x, w) dw, for i, j = 1, 2, we obtain the formula (4.10). □

Under the conditions (A) and (B), there exists an invariant measure (\overline{γ}_{1}, \overline{γ}_{2}) for the Markov chain of types (τ_{t}). Analogously to Todorova (1997), Appendix, we obtain that the measure \overline{Γ} = (\overline{γ}_{1}Λ, \overline{γ}_{2}Λ) is invariant for the semigroup U:

⟨Λ, U(t)Φ⟩ = ⟨Λ, Φ⟩, Φ ∈ (S(R^{d}))^{2}.

In this case, from the proof of the above lemma we obtain a more simple expression for the covariance functional.

Corollary 4.2. For Γ = \overline{Γ}, 0 ≤ s ≤ t, the covariance \overline{K} of X is given by

K(s, φ; t, ψ) = ∑_{i,j=1}^{2} ∫ \overline{γ}_{i} [φ(x) ∫ p_{t−s}^{(i)}(x, y)ψ(y) dy] (x) dx

+ ∑_{j,l=1}^{2} ∫_{0}^{s} ∫ a_{jl} [φ(y) ∫ p_{s−v}^{(j)}(x, y) dy ∫ ψ(w)p_{t−v}^{(l)}(x, w) dw] (x) dx dv,

where a_{jl} = ∑_{i=1}^{2} \overline{γ}_{i}V_{i}m_{i}^{(2)}(j, l).

Substituting K(s, φ; t, ψ) in (2.2), first for Φ, Ψ of the form Φ = φ ⊗ ψ, Ψ = \overline{φ} ⊗ \overline{ψ}, and then for Φ, Ψ of the general form, and using the linearity of the tensor product, we obtain analogously to Lemma 5.3 in Gorostiza and Todorova (1998) the following result.

Lemma 4.3. For s ≤ u, r ≤ v we have

J_{s,r,u,v}(Φ, \overline{Φ}) = J_{s,r,u,v}^{(1)}(Φ, \overline{Φ}) + J_{s,r,u,v}^{(2)}(Φ, \overline{Φ}),

where

J_{s,r,u,v}^{(1)}(Φ, \overline{Φ})

= ∑_{i,j=1}^{2} ∫ U_{i}(s)U_{j}(r) (∫ Φ(x, z)\overline{Φ}(y, w)p_{u−s}^{(i)}(x, y)p_{v−r}^{(j)}(z, w) dy dw) dx dz
+ ∑_{i,p,q=1}^{2} ∫_{0}^{r} ∫ U_{i}(s)L_{pq}(\overline{ω})
× (∫ Φ(x, \overline{y})\overline{Φ}(y, \overline{w})p_{u−s}^{(i)}(x, y)p_{r−\overline{ω}}^{(p)}(\overline{x}, \overline{y})p_{v−\overline{ω}}^{(q)}(\overline{x}, \overline{w}) dy d\overline{y} d\overline{w}) dx d\overline{x} d\overline{ω}
+ ∑_{i,j,p,q=1}^{2} ∫_{0}^{s} ∫_{0}^{r} ∫ L_{ji}(ω)L_{pq}(\overline{ω})
× (∫ Φ(y, \overline{y})\overline{Φ}(w, \overline{w})p_{s−ω}^{(j)}(x, y)p_{u−ω}^{(i)}(x, w)p_{r−\overline{ω}}^{(p)}(\overline{x}, \overline{y})p_{v−\overline{ω}}^{(q)}(\overline{x}, \overline{w}) dy dw d\overline{y} d\overline{w})
× d\overline{x} dx d\overline{ω} dω,

and

J_{s,r,u,v}^{(2)}(Φ, \overline{Φ}) = J_{s,r,u,v}^{(1)}(Φ, S\overline{Φ}),

where SΦ(x, y) = Φ(y, x).

Corollary 4.4. For Γ = \overline{Γ}, we have

\overline{J}_{s,r,u,v}^{(1)}(Φ, \overline{Φ}) = \overline{J}_{(1),s,r,u,v}(Φ, \overline{Φ}) + \overline{J}_{(2),s,r,u,v}(Φ, \overline{Φ}) + \overline{J}_{(3),s,r,u,v}(Φ, \overline{Φ}),

where

\overline{J}_{(1),s,r,u,v}(Φ, \overline{Φ}) = ∑_{i,j=1}^{2} ∫ \overline{γ}_{i}\overline{γ}_{j} (∫ Φ(x, z)\overline{Φ}(y, w)p_{u−s}^{(i)}(x, y)
× p_{v−r}^{(j)}(z, w) dy dw) dx dz  (4.11)
\overline{J}_{(2),s,r,u,v}(Φ, \overline{Φ}) = ∑_{i,p,q=1}^{2} ∫_{0}^{r} ∫ \overline{γ}_{i}a_{pq} (∫ Φ(x, \overline{y})\overline{Φ}(y, \overline{w})
× p_{u−s}^{(i)}(x, y)p_{r−\overline{ω}}^{(p)}(\overline{x}, \overline{y})p_{v−\overline{ω}}^{(q)}(\overline{x}, \overline{w}) dy d\overline{y} d\overline{w}) dx d\overline{x} d\overline{ω} (4.12)
\overline{J}_{(3),s,r,u,v}(Φ, \overline{Φ}) = ∑_{i,j,p,q=1}^{2} ∫_{0}^{s} ∫_{0}^{r} ∫ a_{ji}a_{pq} (∫ Φ(y, \overline{y})\overline{Φ}(w, \overline{w})
× p_{s−ω}^{(j)}(x, y)p_{u−ω}^{(i)}(x, w)p_{r−\overline{ω}}^{(p)}(\overline{x}, \overline{y})p_{v−\overline{ω}}^{(q)}(\overline{x}, \overline{w})
× dy dw d\overline{y} d\overline{w}) d\overline{x} dx d\overline{ω} dω,  (4.13)

and

\overline{J}_{s,r,u,v}^{(2)}(Φ, \overline{Φ}) = \overline{J}_{s,r,u,v}^{(1)}(Φ, S\overline{Φ}).

Having the expression for the function J_{s,r,u,v}, condition (1) of Theorem 2.2 is proved analogously to the Corollary 5.7 of Gorostiza and Todorova (1998).

Let q_{t}^{(i)}(x, y), i = 1, 2, denote the transition density of the symmetric α_{i}-stable motion. Using a renewal argument (see e.g. Athreya and Ney, 1972), conditioning on the first

branching and using that the probability of a jump from type i to type j ≠ i, for the Markov chain of types is −A_{ij}/A_{ii}, i, j = 1, 2, (see e.g. Grimmett and Stirzaker, 1982, Proposition 6.9.14), we obtain the following lemma.

Lemma 4.5. The transition densities of the basic process satisfy the following system of equations:

p_{t}^{(1)}(x, y) = e^{−V_{1}t}q_{t}^{(1)}(x, y) + A_{1}V_{1} ∫_{0}^{t} e^{−V_{1}s} ∫ q_{s}^{(1)}(x, z)p_{t−s}^{(2)}(z, y) dz ds,
p_{t}^{(2)}(x, y) = e^{−V_{2}t}q_{t}^{(2)}(x, y) + A_{2}V_{2} ∫_{0}^{t} e^{−V_{2}s} ∫ q_{s}^{(2)}(x, z)p_{t−s}^{(1)}(z, y) dz ds, (4.14)

where A_{1} = −A_{12}V_{1}/A_{11}, A_{2} = −A_{21}V_{2}/A_{22}. In particular, p_{t}^{(i)}(x, y) ≥ Kq_{t}^{(i)}(x, y), x, y ∈ R^{d}, i = 1, 2, where K = e^{− max{V_{1},V_{2}}}.

Analogously to Lemma 5.9 of Gorostiza and Todorova (1998), we prove the following result.

Lemma 4.6. (a) The densities p_{t}^{(i)}(x, y), i = 1, 2, satisfy the inequalities

p_{t}^{(i)}(x, y) ≥ aq_{t}^{(i)}(x, y), x, y ∈ R^{d},  (4.15)
p_{t}^{(i)}(x, y) ≤ bt^{−d/ min{α_{1},α_{2}}}, x, y ∈ R^{d},  (4.16)

for some positive constants a and b.

(b) p_{t}^{(i)}(x, y) = p_{t}^{(i)}(x − y), i = 1, 2.

(c) For i ≠ j,

p_{t}^{(i)}(x, y) ≤ q_{t}^{(i)}(x, y) + ∫_{0}^{t} ∫ q_{s}^{(i)}(x, z)q_{t−s}^{(j)}(z, y) dz μ_{t}^{(i)}(ds), t ∈ [0, 1],

where

μ_{t}^{(i)}(ds) = P (θ_{t}^{(i)} ∈ ds)

and θ_{t}^{(i)} the total time in the interval [0,t) during which the basic process moves according to α_{i}, starting with type i, on the event that there is a change of type in (0, t].

(d) There is a positive constant C = C(d, α_{1}, α_{2}) such that

sup_{x,z∈R^{d}} ∫ q_{r}^{(1)}(x, z)q_{s}^{(2)}(z, y) dz ≤ C(r + s)^{−d/ min{α_{1},α_{2}}}

for all s, r ∈ [0, 1].

Remark. If we consider in (d) densities q^{(·)} of the same type, the inequality holds by the Chapman-Kolmogorov formula and the self-similarity. In the case of two types the inequality plays the role of Chapman-Kolmogorov and self-similarity.

We can now prove the condition (2) of Theorem 2.2 analogously to the Proposition 5.10 of Gorostiza and Todorova (1998), obtaining the following result.

Lemma 4.7. The limit lim_{ε,δ→0} J_{s,r,u,v}(Φ_{ε,φ}^{f}, Φ_{δ,φ}^{g}) exists for all f, g ∈ F, φ ∈ S(R^{d}), s, r, u, v ∈ [0, 1], and this limit does not depend on f, g. In the case of the invariant measure we have

lim_{ε,δ→0} J_{s,r,u,v}^{(1)}(Φ_{ε,φ}^{f}, Φ_{δ,φ}^{g})

= ∑_{i,j=1}^{2} ∫ \overline{γ}_{i}\overline{γ}_{j}φ(x)φ(y)p_{u−s}^{(i)}(x, y)p_{v−r}^{(j)}(x, y) dx dy

+ ∑_{i,p,q=1}^{2} ∫_{0}^{r} ∫ \overline{γ}_{i}a_{pq}φ(x)φ(y)p_{u−s}^{(i)}(x, y)p_{r−\overline{ω}}^{(p)}(\overline{x}, x)p_{v−\overline{ω}}^{(q)}(\overline{x}, y) dy dx d\overline{x}d\overline{ω}

+ ∑_{i,j,p,q=1}^{2} ∫_{0}^{s} ∫_{0}^{r} ∫ a_{ji}a_{pq} (∫ φ(y)φ(w)p_{s−ω}^{(j)}(x, y)p_{u−ω}^{(i)}(x, w)p_{r−\overline{ω}}^{(p)}(\overline{x}, y)
× p_{v−\overline{ω}}^{(q)}(\overline{x}, w)dy dwd\overline{x} dx) d\overline{ω} dω.  (4.17)

In the following lemma we obtain the covariance inequalities for the process X in the stationary and the general case, and for the functions J in both cases. The proof is analogous to the proof of Lemma 5.11 of Gorostiza and Todorova (1998).

Lemma 4.8. (a) There exist positive constants c_{1} and c_{2} such that for any positive φ, ψ ∈ S(R^{d}),

c_{1}\overline{K}(s, φ; t, ψ) ≤ K(s, φ; t, ψ) ≤ c_{2}\overline{K}(s, φ; t, ψ).

(b) There exist positive constants b_{1} and b_{2} such that for any positive Φ^{(1)}, Φ^{(2)} ∈ S(R^{2d}),

b_{1}\overline{J}_{s,r,u,v}^{(i)}(Φ^{(1)}, Φ^{(2)}) ≤ J_{s,r,u,v}^{(i)}(Φ^{(1)}, Φ^{(2)}) ≤ b_{2}\overline{J}_{s,r,u,v}^{(i)}(Φ^{(1)}, Φ^{(2)}), i = 1, 2.

With these preliminary results we now proceed to prove Theorem 3.2.

Proof of Theorem 3.2.

(a) Case d ≥ 2 min{α_{1}, α_{2}}.

We assume α_{1} ≤ α_{2}. Let φ > 0. Using the Fatou lemma, Lemma 4.8 (b) and Corollary 4.4 we obtain

lim inf_{ε→0} ∫_{[0,1]^{4}} J_{s,r,u,v}(Φ_{ε,φ}^{f}, Φ_{ε,φ}^{f}) ds dr du dv

≥ b_{1} lim inf_{ε→0} ∫_{[0,1]^{4}} \overline{J}_{s,r,u,v}^{(1)}(Φ_{ε,φ}^{f}, Φ_{ε,φ}^{f}) ds dr du dv

≥ b_{1}\overline{γ}_{1}^{2} ∫_{[0,1]^{4}} ∫ p_{|u−s|}^{(1)}(x, y)p_{|v−r|}^{(1)}(x, y)φ(x)φ(y) dx dy ds dr du dv

+b_{1}\overline{γ}_{1}^{2} ∫_{[0,1]^{4}} ∫ p_{|u−r|}^{(1)}(x, y)p_{|v−s|}^{(1)}(x, y)φ(x)φ(y) dx dy ds dr du dv.

But the last integral is ∞ (see Bojdecki and Gorostiza, 1995, Lemma 2.6 a)).

Using the part 2 of Theorem 2.2 we conclude that in this case the SILT does not exist.

(b) Case d < 2 min{α_{1}, α_{2}}.

We suppose α_{1} ≤ α_{2}. We will verify conditions (iii) and (iv) of Theorem 2.2.

From the positivity of the operators U_{i} and L_{ij} and from Lemma 4.8 (b)

|J_{s,r,u,v}^{(1)}(Φ_{ε,φ}^{f}, Φ_{δ,φ}^{g})| ≤ J_{s,r,u,v}^{(1)}(|Φ_{ε,φ}^{f}|, |Φ_{δ,φ}^{g}|) ≤ b_{2}\overline{J}_{s,r,u,v}^{(1)}(|Φ_{ε,φ}^{f}|, |Φ_{δ,φ}^{g}|) = b_{2}\overline{J}_{s,r,u,v}^{(1)}(Φ_{ε,|φ|}^{f}, Φ_{δ,|φ|}^{g}).

Therefore it suffices to bound the terms of \overline{J}_{s,r,u,v}^{(1)} in (4.12) and (4.13). From Gorostiza and Todorova (1998) we have a bound G_{φ}(s, r, u, v) for (4.11) which satisfies the conditions (iii) and (iv) of Theorem 2.2.

For (4.12) we can find a bound G_{φ}(s, r, u, v) which satisfies the conditions (iii) and (iv) of Theorem 2.2 proceeding as follows: first we apply the inequality of Lemma 4.6 (c) to the densities p_{t}^{(i)} and then for the obtained terms we apply the inequality of Lemma 4.6 (d) to both factors q_{t}^{(i)} which contain the variable \overline{x} in (4.12). For instance, applying the inequalities

p_{|u−s|}^{(1)}(x, y) ≤ q_{|u−s|}^{(1)}(x, y) + ∫_{0}^{|u−s|} ∫ q_{m}^{(1)}(x, z)q_{|u−s|−m}^{(2)}(z, y) dz μ_{|u−s|}^{(1)}(ds) (4.18)
p_{r−\overline{ω}}^{(p)}(\overline{x}, \overline{y}) ≤ q_{r−\overline{ω}}^{(p)}(\overline{x}, \overline{y}) + ∫_{0}^{r−\overline{ω}} ∫ q_{n}^{(p)}(\overline{x}, \overline{z})q_{r−\overline{ω}−n}^{(p′)}(\overline{z}, \overline{y}) d\overline{z} μ_{r−\overline{ω}}^{(p)}(ds)  (4.19)
p_{v−\overline{ω}}^{(q)}(\overline{x}, \overline{w}) ≤ q_{v−\overline{ω}}^{(q)}(\overline{x}, \overline{w}) + ∫_{0}^{v−\overline{ω}} ∫ q_{l}^{(q)}(\overline{x}, \overline{t})q_{v−\overline{ω}−l}^{(q′)}(\overline{t}, \overline{w}) d\overline{t} μ_{v−\overline{ω}}^{(q)}(ds)  (4.20)

in (4.12) and grouping the second term in (4.18) and the first terms in (4.19) and (4.20), we obtain the following term, for which we will obtain a bound G_{φ}^{(1)}(s, r, u, v):

∫_{0}^{|u−s|} ∫_{0}^{r} |φ(x)||φ(y)|f_{ε}(x − \overline{y})g_{δ}(y − \overline{w})q_{m}^{(1)}(x, z)q_{|u−s|−m}^{(2)}(z, y)q_{r−\overline{ω}}^{(p)}(\overline{x}, \overline{y})

× q_{|v−\overline{ω}|}^{(q)}(\overline{x}, \overline{w})dx dy dz d\overline{x} d\overline{y} dw d\overline{ω} μ_{|u−s|}^{(1)}(dm)
≤ sup_{x} |φ(x)| ∫_{0}^{r}(r − \overline{ω} + |v − \overline{ω}|)^{−d/α_{1}} d\overline{ω} ∫_{0}^{|u−s|} |φ(y)|f_{ε}(x − \overline{y})g_{δ}(y − \overline{w})
× q_{m}^{(1)}(x, z)q_{|u−s|−m}^{(2)}(z, y)d\overline{w} d\overline{y} dx dz dy μ_{|u−s|}^{(1)}(dm)
= sup_{x} |φ(x)| ∫ |φ(y)| dy ∫_{0}^{r}(r − \overline{ω} + |v − \overline{ω}|)^{−d/α_{1}} d\overline{ω}

(the last equality is obtained by multiple integration in the indicated order, and using that q_{t}^{(i)} are densities).

Using x^{2} + y^{2} ≥ 2xy, x, y ≥ 0, and \overline{ω} ∈ [0, r] we obtain for d < 2α_{1},

∫_{0}^{r}(r − \overline{ω} + |v − \overline{ω}|)^{−d/α_{1}} d\overline{ω} ≤ |v − r|^{−d/2α_{1}} ∫_{0}^{r}(r − \overline{ω})^{−d/2α_{1}} d\overline{ω} ≤ c(|v − r|)^{−d/2α_{1}},

where c is a constant. The function G_{φ}^{(1)}(s, r, u, v) = c sup_{x} |φ(x)|(∫ |φ(y)| d)(|v − r|)^{−d/2α_{1}} satisfies the conditions (iii) and (iv) of Theorem 2.2.

We bound analogously the other terms of (4.12) and those of (4.13).

We proceed analogously with the terms of \overline{J}_{s,r,u,v}^{(2)}.

This completes the proof of Theorem 3.2. □. Acknowledgment. Most of this work was done during my stay at Centro de Investi-

gación y de Estudios Avanzados, which I thank for the support I received.

References

R. J. Adler, R. Feldman and M. Lewin, Intersection local times for infinite systems of Brownian motions and for the Brownian density process, Ann. Probab. 19 (1991), 192-220.

R. J. Adler and J. S. Rosen, Intersection local times of all orders for Brownian and stable density processes -construction, renormalization and limit laws, Ann. Probab. 21 (1993), 1073-1123.

K. B. Athreya and P. E. Ney, Branching Processes, Springer-Verlag, 1972.

T. Bojdecki and L. G. Gorostiza, Self-intersection local time for Gaussian S′(R^{d})- processes: Existence, path continuity and examples, Stoch. Proc. Appl. 60 (1995), 191-226.

T. Bojdecki and L. G. Gorostiza, Self-intersection local time for Gaussian S′-processes, and application to fluctuation limits of branching particle systems, in “Stochastic Analysis:

Random Fields and Measure-Valued Processes”, Israel Mathematical Conference Proceedings 10 (1996), 49-55.

T. Bojdecki and L. G. Gorostiza, Self-intersection local time for S′(R^{d})-Wiener processes and related Ornstein-Uhlenbeck processes (preprint), (1997).

B. Fernández, Markov properties of the fluctuation limit of a particle system with death, J. Math. Anal. Appl. 155 (1991), 66-77.

L. G. Gorostiza and E. Todorova, Self-intersection local time of an S′(R^{d})-valued process involving motions of two types, (1998), preprint.

G. Grimmet and D. Stirzaker, Probability and Random Processes, Oxford, (1982).

J. A. López-Mimbela, Fluctuation limits of multitype branching random fields, J. Multiv. Anal. 40 (1992), 56-83.

E. Todorova, Tiempo local de autointersección del proceso de densidad de un sistema de partículas de dos tipos, Ph.D. thesis, Centro de Investigación y de Estudios Avanzados, (1997), Mexico.