Georgian Mathematical Journal Volume 8 (2001), Number 2, 389–400

ON STOCHASTIC DIFFERENTIAL EQUATIONS IN A CONFIGURATION SPACE

A. SKOROKHOD

Abstract. Infinite systems of stochastic differential equations for randomly perturbed particle systems with pairwise interaction are considered. It is proved that under some reasonable assumption on the potential function there exists a local weak solution to the system and it is weakly locally unique for a wide class of initial conditions.

2000 Mathematics Subject Classification: 60H10, 60H30.

Key words and phrases: Configuration space, stochastic differential equation, local weak solution.

1. Introduction

We consider a sequence of R^{d}-valued stochastic processes

x_{k}(t), k = 1, 2, …,

satisfying the system of stochastic differential equations of the form

dx_{k}(t) = ∑_{i≠k} a(x_{k}(t) − x_{i}(t))dt + σdw_{k}(t), k = 1, 2, …,  (1)

where a(x) = −U_{x}(x), and U : R^{d} → R is a smooth function for |x| > 0, and σ > 0 is a constant, w_{k}(t), k = 1, 2, …, is a sequence of independent Wiener processes in R^{d}. System (1) describes the evolution of systems of pairwise interacting particles with the pairwise potential U(x) which is perturbed by Wiener noises. The problem is to find conditions under which the system has a solution and this solution is unique.

Unperturbed systems were considered by many authors. We notice the recent articles of S. Albeverio, Yu. G. Kondratiev, and M. Röckner [1], [2] where a new powerful method for the investigation of unperturbed systems is proposed. Finite-dimensional perturbed systems were considered in my book [3] and my article [4]. The first general theorem on the existence and uniqueness of the solutions to infinite dimensional stochastic differential equations were obtained by Yu. L. Daletskii in [5]; he considered equations with smooth coefficients in a Hilbert space. The existence and uniqueness of the solution to system (1) for locally bounded smooth potentials and d ≤ 2 was proved by J. Fritz in [6]. The main result of this article was published in [7] without a complete proof.

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2. The Space Γ

It is convenient to consider system (1) in the configuration space Γ which is the set of locally finite counting measures γ on the Borel σ-algebra B(R^{d}) of the space R^{d}. So a measure γ ∈ Γ satisfies the condition: the support S_{γ} of the measure γ is a sequence of different points {x_{k}, k ∈ N } of R^{d} for which

|x_{k}| → ∞, γ(A) = ∑_{k} 1_{A}(x_{k}).

The topology in Γ is generated by the weak convergence of measures: γ_{n} → γ_{0} if

∫ ϕ(x)γ_{n}(dx) → ∫ ϕ(x)γ_{0}(dx)

for ϕ ∈ C_{f} where C_{f} is the set of continuous functions ϕ : R^{d} → R with bounded supports.

We use the notation

⟨ϕ, γ⟩ = ∫ ϕ(x)γ(dx), ϕ ∈ C_{f},

and

⟨Φ, γ × γ⟩ = ∫ Φ(x, x′)γ(dx)γ(dx′) − ∫ Φ(x, x)γ(dx),

where Φ : (R^{d})^{2} → R is a continuous function with a bounded support. We rewrite system (1) for Γ-valued function γ_{t} for which

⟨ϕ, γ⟩ = ∑_{k} ϕ(x_{k}(t)), ϕ ∈ C_{f}.

Using the Itô’s formula, and considering the function a as a function of two variables a(x − x′), we obtain the relation

d⟨ϕ, γ_{t}⟩ = ⟨(ϕ′, a), γ_{t} × γ_{t}⟩dt + \frac{σ^{2}}{2} ⟨Δϕ, γ_{t}⟩
+ ∑_{k} σ(ϕ′(x_{k}(t)), dw_{k}(t)), ϕ ∈ C_{f}^{(2)},  (2)

where Δϕ(x) = Trϕ′′(x), and C_{f}^{(2)} is the set of ϕ ∈ C_{f} for which ϕ′(x) and ϕ′′(x) are continuous bounded functions.

A weak solution to equation (1). A Γ-valued stochastic process γ_{t}(ω) is called a weak solution to system (2) if, for all ϕ ∈ C_{f}^{(2)}, the stochastic process

μ_{ϕ}(ω, t) = ⟨ϕ, γ_{t}⟩ − ∫_{0}^{t}[⟨(ϕ′, a), γ_{s} × γ_{s}⟩ + \frac{σ^{2}}{2} ⟨Δϕ, γ_{s}⟩]ds  (3)

is a martingale with respect to the filtration (F_{t})_{t≥0}, where F_{t} = σ(γ_{s}, s ≤ t), and the square characteristic of the martingale is

⟨μ_{ϕ}, μ_{ϕ}⟩_{t} = σ^{2} ∫_{0}^{t}⟨(ϕ′, ϕ′), γ_{s}⟩ds.  (4)

If γ_{t}(ω) is a weak solution to system (2) and

⟨ϕ, γ_{t}(ω)⟩ = ∑ ϕ(x_{k}(t))

for all ϕ ∈ C_{f}, then the sequence {x_{k}(t), k ∈ N } is a weak solution to system (1).

Weak uniqueness. Let γ_{0} ∈ Γ. System (2) has a unique weak solution with the initial value γ_{0} if for any pair of weak solutions to system (2) γ_{t}^{1}(ω) and γ_{t}^{2}(ω) satisfying the condition γ_{0}^{1} = γ_{0}^{2} = γ_{0}, the following relations are fulfilled:

EΦ(ξ_{11}^{1}, …, ξ_{1m}^{1}, …, ξ_{l1}^{1}, …, ξ_{lm}^{1}) = EΦ(ξ_{11}^{2}, …, ξ_{1m}^{2}, …, ξ_{l1}^{2}, …, ξ_{lm}^{2}), (5)

where Φ(y_{11}, …, y_{lm}) is a continuous bounded function on R^{lm} and,

ξ_{ij}^{k} = ⟨ϕ_{i}, γ_{t_{j}}^{k}(ω)⟩, k = 1, 2, i = 1, …, l, j = 1, …, m,
ϕ ∈ C_{f}, t_{1}, …, t_{m} ∈ R_{+}.

This means that the distribution of the stochastic process γ_{t}^{k} does not depend on k, i.e., the distribution of the weak solution to system (2) is unique if it exists.

Local weak solutions. Let γ_{t}(ω) be a continuous Γ-valued stochastic process, and (F_{t})_{t≥0} be the filtration generated by it. γ_{t}(ω). is called a local weak solution to system (2) if there exists a stopping time τ with respect to the filtration (F_{t})_{t≥0} for which P {τ > 0} = 1 and the stochastic process μ_{ϕ}(ω, t) which is determined by relation (3) is a martingale for t < τ with the square characteristic given by equality (4).

Local weak uniqueness. Let γ_{0} ∈ Γ. System (2) has a locally unique weak solution with the initial value γ_{0} if, for any pair of locally weak solutions to system (2) γ_{t}^{1} and γ_{t}^{2} satisfying the condition γ_{0}^{1} = γ_{0}^{2} = γ_{0}, relation (5) is fulfilled for

ξ_{ij}^{k} = ⟨ϕ_{i}, γ_{t_{j}}^{k}(ω)⟩I_{{t_{j}<τ^{k}}} ,

where τ^{k}, k = 1, 2, are stopping times with respect to the filtration (F_{t}^{k})_{t≥0} generated by the stochastic process γ_{t}^{k}(ω).

Compacts in Γ. For any γ ∈ Γ and a continuous decreasing function λ(t) : (0, ∞) → R_{+} for which λ(0+) = +∞, λ(+∞) > 0 there exists a continuous decreasing function Φ(t) : [0, ∞) → R_{+} with Φ(+∞) = 0 such that

∫∫ Φ(|x|)Φ(|x′|)λ(|x − x′|)1_{{x≠x′}}γ(dx)γ(dx′) < ∞.  (6)

Denote

Φ_{λ}(x, x′) = Φ(|x|)Φ(|x′|)λ(|x − x′|).  (7)

For any compact set K from Γ and any fanction λ satisfying the conditions mentioned before there exists a function of the form given by relation (7) for which

sup_{γ∈K}⟨Φ_{λ}, γ × γ⟩ < ∞.

Note that the set

{γ : ⟨Φ_{λ}, γ × γ⟩ ≤ c}

is a compact in Γ for any Φ_{λ} of form (7) and c > 0. Denote by Γ_{Φ,λ} the set of those γ ∈ Γ for which relation (6) is fulfilled. Set

d_{Φ,λ}(γ_{1}, γ_{2}) = sup{|⟨ϕΦ, γ_{1}⟩−⟨ϕΦ, γ_{2}⟩ : ϕ ∈ Lip^{1}|}+|⟨Φ_{λ}, γ_{1}×γ_{1}⟩−⟨Φ_{λ}, γ_{2}×γ_{2}⟩|,

where

Lip^{1} = \{ϕ ∈ C_{f} : sup_{x} |ϕ(x)| ≤ 1, sup_{x,x′} \frac{|ϕ(x) − ϕ(x′)|}{|x − x′|} ≤ 1\}.

Γ_{Φ,λ} with the distance d_{Φ,λ} is a separable locally compact space.

3. An Extension of Girsanov’s Formula

Assume that the potential function U(x) satisfies the condition

(PC) U(x) = u(|x|) where the function u : (0, ∞) → R is continuous, it has continuous derivatives u′, u′′, there exists a constant r > 0 for which u(t) = 0 for t > r, and

∫ t^{d−1}|u(t)|dt < ∞.

Free particle processes. Let a measure γ_{0} satisfy the condition (IC)

⟨ϕ_{δ}, γ_{0}⟩ < ∞, ϕ_{δ}(x) = exp {−δ|x|^{2}}.

Introduce Γ-valued stochastic processes by the relation

⟨ϕ, γ_{t}^{∗}(γ_{0}, ω)⟩ = ∑_{k} ϕ(x_{k}^{0} + σw_{k}(t)),

where

∑_{k} ϕ(x_{k}^{0}) = ⟨ϕ, γ_{0}⟩.

It is easy to check that γ_{t}^{∗}(γ_{0}, ω) is a continuous Γ-valued stochastic process if γ_{0} ∈ Γ^{0}, where Γ^{0} is the set of finite measures from Γ. There exist functions Φ, λ for which P {γ_{t}^{∗}(γ_{0}, ω) ∈ Γ_{Φ,λ}} = 1 for all t > 0. The stochastic process γ_{t}^{∗}(γ_{0}, ω) is continuous in the space Γ_{Φ,λ}, and for any t_{0} > 0 the function

EF (γ^{∗}(γ_{0}, ω))

is a continuous function in γ_{0} ∈ Γ_{Φ,λ} if F is a bounded continuous function on C_{[0,t_{0}]}(Γ_{Φ,λ}) which is the space of continuous Γ_{Φ,λ}-valued functions on the interval [0, t_{0}].

Girsanov’s formula for finite systems. Let γ_{0}^{n} be a sequence of finite measures from Γ satisfying the condition:

γ_{0}^{n} → γ_{0}, γ_{0}^{n} ≤ γ_{0}^{n+1}.

It was proved in [4] that under the condition (PC), for any n, there exists a unique strong solution to system (2) with the initial value γ_{0}^{n}. Denote it by γ_{t}(γ_{0}^{n}, ω).

Lemma 1. Set a(x, x′) = a(x − x′),

G_{1}^{n}(t) = σ^{−1} ∑_{x_{i}∈S_{γ^{n}_{0}}} ∫_{0}^{t}(⟨a(x_{i} + σw_{i}(s), .), γ_{s}^{∗}(γ_{0}^{n}, ω)⟩, dw_{i}(s)),

G_{2}^{n}(t) = σ^{−2} ∫_{0}^{t} ∫ |⟨a(x, .), γ_{s}^{∗}(γ_{0}^{n}, ω)⟩|^{2}γ_{s}^{∗}(γ_{0}^{n}, ω, dx)ds,

ρ_{n}(t) = exp \{G_{1}^{n}(t) − \frac{1}{2}G_{2}^{n}(t)\}.

Then

EΦ(⟨ϕ_{1}, γ_{t_{1}}(γ_{0}^{n}, ω)⟩, …, ⟨ϕ_{k}, γ_{t_{k}}(γ_{0}^{n}, ω⟩)
= Eρ_{n}(t)Φ(⟨ϕ_{1}, γ_{t_{1}}^{∗}(γ_{0}^{n}, ω)⟩, …, ⟨ϕ_{k}, γ_{t_{k}}^{∗}(γ_{0}^{n}, ω)⟩)

for all k = 1, 2, …, bounded continuous functions Φ : R^{k} → R and ϕ_{1}, …, ϕ_{k} ∈ C_{f}, t_{1}, …, t_{k} ∈ [0, t].

The proof of the lemma can be obtained using the approximation of the function a(x, x′) by smooth functions since for smooth a(x, x′) the proof is a consequence of Girsanov’s formula [8].

Introduce the stochastic processes

w_{k}^{c}(t) = ∫_{0}^{t} 1_{{|x_{k}+w_{k}(s)|≤c}}dw_{k}(s), x_{k} ∈ S_{γ_{0}^{n}},

where c > 0 is a constant. Let F_{t}^{n,c} be the σ -algebra generated by

{w_{k}^{c}(s), s ≤ t, x_{k} ∈ S_{γ_{0}^{n}}}.

Lemma 2.

E(ρ_{n}/F_{t}^{n,c}) = ρ_{n}(c, t),

where

ρ_{n}(c, t) = exp \{G_{1}^{n}(c, t) − \frac{1}{2}G_{2}^{n}(c, t)\},

and

G_{1}^{n}(c, t) = σ^{−1} ∑_{x_{i}∈S_{γ^{n}_{0}}} ∫_{0}^{t}(E(⟨a(x_{i} + σw_{i}(s), .), γ_{s}^{∗}(γ_{0}^{n}, ω)⟩/F_{s}^{n,c}), dw_{i}^{c}(s)),

G_{2}^{n}(c, t) = σ^{−2} ∫_{0}^{t} ∫ |E(⟨a(x, .), γ_{s}^{∗}(γ_{0}^{n}, ω)⟩/F_{s}^{n,c})|^{2}1_{{|x|≤c}}γ_{s}^{∗}(γ_{0}^{n}, ω, dx)ds.

The proof rests on the statement below.

Statement. Let F_{t}, t ∈ R_{+} be a continuous filtration, and its subfiltration F_{t}^{∗}, t ∈ R_{+}, satisfy the condition

(RE)E(ξ/F_{t}) is a F_{t}^{∗}-measurable random variable if ξ is a bounded F_{∞}^{∗}- measurable random variable.

Let μ_{t} be an F_{t} -martingale with the square characteristic ⟨μ⟩_{t} for which the stochastic process

ρ(t) = exp \{μ_{t} − \frac{1}{2}⟨μ⟩_{t}\}

is a martingale. Then

E(ρ(t)/F_{t}^{∗}) = exp \{μ_{t}^{∗} − \frac{1}{2}⟨μ^{∗}⟩_{t}\},

where μ_{t}^{∗} = E(μ_{t}/F_{t}^{∗}) is a martingale, and ⟨μ^{∗}⟩_{t} is its square characteristic. Proof. Set μ̃_{t} = μ_{t} − μ_{t}^{∗}. Condition (RE) implies the relation

E( ∫_{0}^{t} g(s)dμ̃_{s}/F_{∞}^{∗}) = 0

for all F_{t}-adapted functions g(t) for which

E| ∫_{0}^{t} g(s)dμ̃_{s}| < ∞.

Set

ρ^{∗}(t) = exp \{μ_{t}^{∗} − \frac{1}{2}⟨μ^{∗}⟩_{t}\}, ρ̃(t) = exp \{μ̃_{t} − \frac{1}{2}⟨μ̃⟩_{t}\},

where ⟨μ̃⟩_{t} is the square characteristic of the martingale μ̃_{t}. The proof follows from the relations ρ(t) = ρ^{∗}(t)ρ̃(t) and

E(ρ̃(t) − 1/F_{∞}^{∗}) = E( ∫_{0}^{t} ρ̃(s)dμ̃(s)/F_{∞}^{∗}) = 0.

Remark 1. Assume that ϕ_{i}(x) = 0 for |x| ≥ c, i = 1, 2, …, k, ϕ_{1}, …, ϕ_{k} ∈ C_{f}, t_{1}, …, t_{k} ∈ [0, t]. Then for Φ satisfying the conditions of Lemma 1 we have the relation

EΦ(⟨ϕ_{1}, γ_{t_{1}}(γ_{0}^{n}, ω)⟩, …, ⟨ϕ_{k}, γ_{t_{k}}(γ_{0}^{n}, ω)⟩)
= Eρ_{n}(c, t)Φ(⟨ϕ_{1}, γ_{t_{1}}^{∗}(γ_{0}^{n}, ω)⟩, …, ⟨ϕ_{k}, γ_{t_{k}}^{∗}(γ_{0}^{n}, ω)⟩).

Remark 2. Denote by F_{t}^{c} the σ-algebra generated by {w_{k}^{c}(s), s ≤ t, x_{k} ∈ S_{γ_{0}}}. Then there exist the limits in probability

lim_{n→∞} G_{1}^{n}(c, t) = G_{1}(c, t), lim_{n→∞} G_{2}^{n}(c, t) = G_{2}(c, t),

lim_{n→∞} ρ_{n}(t) = ρ(c, t) = exp \{G_{1}(c, t) − \frac{1}{2}G_{2}(c, t)\},

where

G_{1}(c, t) = σ^{−1} ∑_{x_{i}∈S_{γ_{0}}} ∫_{0}^{t}(E(⟨a(x_{i} + σw_{i}(s), .), γ_{s}^{∗}(γ_{0}, ω)⟩/F_{s}^{c}), dw_{i}^{c}(s))),

G_{2}(c, t) = σ^{−2} ∫_{0}^{t} ∫ |E(⟨a(x, .), γ_{s}^{∗}(γ_{0}, ω)⟩/F_{s}^{c})|^{2}1_{{|x|≤c}}γ_{s}^{∗}(γ_{0}, ω, dx)ds.

ON SDES IN A CONFIGURATION SPACE 395

Let w(t) be a standard Wiener process . Introduce the functions

Q_{c}(s, x, B) = P {x + σw(s) ∈ B, inf_{u≤s} |x + σw(u)| > c},

x ∈ V_{c}, B ∈ B(V_{c}), V_{c} = {x ∈ R_{d} : |x| > c},

Q_{c}^{∗}(s, x, B) = lim_{λ↓1} \frac{Q_{c}(s, x, B)}{Q_{c}(s, λx, V_{c})}, |x| = c, B ∈ B(V_{c}).

Set

θ_{i}(c, s) = inf Δ_{i}(c, s), ζ_{i}(c, s) = sup Δ_{i}(c, s),

where

Δ_{i}(c, s) = {u ≤ s : |x_{i}^{∗}(u)| ≤ c}, x_{i}^{∗}(u) = x_{i} + w_{i}(u).

Then the following statement holds.

Lemma 3.

E(a(x, x_{i}^{∗}(s))1_{{|x_{i}^{∗}(s)|≤c}}/F_{s}^{c}) = a(x, x_{i}^{∗}(s))1_{{|x_{i}^{∗}(s)|≤c}}
+1_{{θ_{i}(c,s)<∞}} ∫ a(x, z)Q_{c}^{∗}(s − ζ_{i}(c, s), x_{i}^{∗}(ζ_{i}(c, s)), dz)
+1_{{θ_{i}(c,s)=+∞}} ∫ a(x, z)Q_{c}(s, x_{i}, dz).

Corollary 1. Introduce the measures

ν_{s}(A) = ∑_{k} 1_{{x_{k}∈A}}1_{{θ_{k}(c,s)=+∞}}, A ∈ B(V_{c}),

ν_{s}^{∗}(Λ, A) = ∑_{k} 1_{{ζ_{k}(c,s)∈Λ}}1_{{x_{k}(ζ_{k}(c,s))∈A}},

Λ ∈ B([0, s]), A ∈ B(V′_{c}), V′_{c} = {x ∈ R_{d} : |x| = c}.

Then

E(⟨a(x, ·), γ_{s}^{∗}(γ_{0}, ω)⟩/F_{s}^{c}) = a_{c}(s, x, ω) + ∫ a(x, x′)1_{{|x′|≤c}}γ_{s}^{∗}(γ_{0}, ω, dx′), where

a_{c}(s, x, ω) = ∫∫ a(x, z)Q_{c}(s, x′, dz)ν_{s}(dx′)
+ ∫ ∫ ∫ a(x, z)Q_{c}^{∗}(s − u, x′, dz)ν_{s}^{∗}(du, dx′).

Corollary 2. The functions G_{k}(c, t), k = 1, 2, can be represented in the form

G_{1}(c, t) = σ^{−1} ∫_{0}^{t} ∑_{i}(a_{c}(s, x_{i}^{∗}(s), ω), dw_{i}^{c}(s)),

and

G_{2}(c, t) = σ^{−2} ∫_{0}^{t} H(c, s)ds,

where

H_{c}(s) = ∑_{i} | ∑_{j≠i} a(x_{i}^{∗}(s), x_{j}^{∗}(s))1_{{|x_{j}^{∗}(s)|≤c}} + a_{c}(s, x_{i}^{∗}(s), ω)|^{2}1_{{|x_{i}^{∗}(s)|≤c}}.

Remark 3. Denote by (F_{t}^{c}(i)) the filtration generated by the stochastic process w_{i}^{c}(t), and by (F_{t}^{c}(i, j)) the filtration generated by the pair of stochastic processes (w_{i}^{c}(t); w_{j}^{c}(t)), i ≠ j. Set

ρ^{i}(c, t) = E(ρ(c, t)/F_{t}^{c}(i)), ρ^{i,j}(c, t) = E(ρ(c, t)/F_{t}^{c}(i, j).

Then

ρ^{i}(c, t) = exp \{ ∫_{0}^{t}(g^{i}(c, s), dw_{i}^{c}(s)) − \frac{1}{2} ∫_{0}^{t} |g^{i}(c, s)|^{2}ds\},

where

σg^{i}(c, s) = [E(a_{c}(s, x) + ∑_{k≠i} a(x, x_{k}^{∗}(s))1_{{|x_{k}^{∗}(s)|≤c}})]_{{x=x_{i}^{∗}(s)}},

and

ρ^{i,j}(c, t) = exp { ∫_{0}^{t}[(g_{ij}^{i}(c, s), dw_{i}^{c}(s)) + (g_{ij}^{j}(c, s), dw_{j}^{c}(s))]}

= exp \{ − \frac{1}{2} ∫_{0}^{t}[|g_{ij}^{i}(c, s)|^{2} + |g_{ij}^{j}(c, s)|^{2}]ds\},

where

σg_{ij}^{i}(c, s) = a(x_{i}^{∗}(s), x_{j}^{∗}(s))1_{{|x_{i}^{∗}(s)|∧|x_{j}^{∗}(s)|≤c}}
+[E(a_{c}(s, x) + ∑ 1_{{k≠i}}1_{{k≠j}}a(x, x_{k}^{∗}(s))1_{{|x_{k}^{∗}(s)|≤c}})]_{{x=x_{i}^{∗}(s)}}.

Remark 4. Assume that τ_{c} is a stopping time with respect to the filtration (F_{t}^{c})_{t≥0} satisfying the condition G_{2}(c, τ_{c}) ≤ c_{1}, where c_{1} is a constant. Then Eρ(c, τ_{c}) = 1 and E(ρ(c, τ_{c}))^{2} ≤ exp {c_{1}}.

Introduce the stochastic processes

w_{i}^{∗}(c, t) = w_{i}(t) − σ^{−1} ∫_{0}^{t∧τ_{c}}a_{c}(s, x_{i}^{∗}(s), ω)1_{{|x_{i}^{∗}(s)|≤c}}ds

−σ^{−1} ∫_{0}^{t∧τ_{c}}∑_{i≠j} a(x_{i}^{∗}(s), x_{j}^{∗}(s))1_{{|x_{i}^{∗}(s)|∨|x_{j}^{∗}(s)|≤c}}ds.  (8)

Lemma 4. Denote by {Ω, F, P }, the probability space generated by the sequence {w_{k}(t), k = 1, 2, … } and let P_{c} be the measure on {Ω, F} for which

\frac{dP_{c}}{dP} (ω) = ρ(c, τ_{c}).

Then {w_{k}(c, t), k = 1, 2, … } is the sequence of independent Wiener processes on the probability space {Ω, F.P_{c}}.

The proof is a consequence of Girsanov’s results [8].

Remark 5. Let c_{1} < c_{2} and τ_{c_{k}} be stopping time with respect to the filtration (F_{t}^{c_{k}})_{t≥0}, k = 1, 2, τ_{c_{1}} < τ_{c_{2}} and G_{2}(c_{1}, τ_{c_{1}}) + G_{2}(c_{2}, τ_{c_{2}}) ≤ c_{3}, where c_{3} is a constant. Then

E(ρ(c_{2}, τ_{c_{2}})/F_{τ_{c_{1}}}^{c_{1}}) = ρ(c_{1}, τ_{c_{1}}).

This formula is a consequence of the relation

E(G_{1}(c_{2}, τ_{c_{2}})/F_{τ_{c_{1}}}^{c_{1}}) = G_{1}(c_{1}, τ_{c_{1}}).

Lemma 5. Let {c_{k}, k = 1, 2, … } be a sequence of positive numbers, for which lim_{k→+∞} c_{k} = +∞.Then there exists a sequence of positive numbers {a_{k}} for which

P { ∑_{k} a_{k}G_{2}(c_{k}, t) < ∞} = 1

for all t > 0.

Proof. Choose a_{k} satisfying the inequality

P (G_{2}(c_{k}, k) > (k^{2}a_{k})^{−1}) < k^{−2}.

Then for any t > 0 we have the relation

∑_{k} P (a_{k}G_{2}(c_{k}, t) > k^{−2}) < t + ∑_{k≥t} k^{−2} < ∞.

This completes the proof.

Corollary 3. Let a sequence {a_{k}} satisfy the statement of Lemma 5. Set

G(t) = ∑_{k} a_{k}G_{2}(c_{k}, t).

With probability 1 G(t) is an increasing continuous function satisfying the relations G(0) = 0, lim_{t→∞} G(t) = ∞. Set

τ^{∗} = inf{t : G(t) > b},  (9)

where b is a positive number. Then

Eρ(c_{k}, τ^{∗}) = 1, E(ρ(c_{k}, τ^{∗}))^{2} ≤ exp {ba_{k}^{−1}}.

4. A Theorem on Existence of a Local Weak Solution

Theorem 1. Let conditions (PC) and (IC) be fulfillrd,and let τ^{∗} be a stopping time introduced by relation (9). Then the following statements hold:

(i) there exists a probability measure P^{∗} on (Ω, F) for which

P^{∗}(A) = lim_{c→∞} E1_{A}ρ(c, τ∗), A ∈ ⋁_{k} F_{τ^{∗}}^{c_{k}}, P^{∗}(A) = P (A), A ∈ F^{∗},

where the σ-algebra F^{∗} is generated by the processes

{x_{k}^{∗}(t ∨ τ^{∗}) − x_{k}^{∗}(t), t ≥ 0, k = 1, 2, … }.

(ii) the stochastic processes given by the formula

σw_{k}^{∗}(t) = x_{k}^{∗}(t) − ∫_{0}^{t∧τ^{∗}}∑_{i≠k} a(x_{k}^{∗}(s), x_{i}^{∗}(s))ds, k = 1, 2, …,

are independent Wiener processes with respect to the filtration (F_{t})_{(t≥0)} on the probability space (Ω, F, P^{∗}).

Proof. Since τ^{∗} is a stopping time, for A ∈ F_{τ^{∗}}^{c_{k}} we can write, using Remark 5, the relations

E1_{A}ρ(c, τ^{∗}) = E1_{A}ρ(c_{k}, τ^{∗})

if c > c_{k}. This implies the existence of limits

lim_{c→∞} 1_{A×B}ρ(c, τ^{∗}), A ∈ ⋁_{k} F_{τ^{∗}}^{c_{k}}, B ∈ F^{∗}.

It follows from the last formula that there exists a limit

lim_{c→∞} Φ(ξ_{1}, …, ξ_{m})ρ(c, τ^{∗}),  (10)

where Φ ∈ C(R^{m}) and

ξ_{i} = ⟨f_{i}, γ_{t_{i}}^{∗}(γ_{0}, ω)⟩, i = 1, …, m,

where f_{i} ∈ C(R^{d}), f_{i}(x) = 0 for |x| large enough.

Now we prove that there exists a probability measure P^{∗} on (Ω, F) for which a limit in the formula (10) is represented in the form E^{∗}Φ(ξ_{1}, …, ξ_{m}), where E^{∗} is the expectation with respect to the probability P^{∗}. Set

f_{n}(x, x′) = (1 − \frac{|x|}{c_{n}})(1 − \frac{|x′|}{c_{n}})(1 + \frac{1}{|x − x′|}) ∨ 0.

Using Corollary 3 we can write the inequality

E⟨f_{n}, γ_{t}^{∗}(γ_{0}, ω) × γ_{t}^{∗}(γ_{0}, ω)⟩ρ(c, τ^{∗})
≤ (E⟨f_{n}, γ_{t}^{∗}(γ_{0}, ω) × γ_{t}^{∗}(γ_{0}, ω)⟩^{2})^{\frac{1}{2}} exp \{\frac{1}{2}ba_{n}^{−1}\}.  (11)

Denote

A_{n}(t) = E⟨f_{n}, γ_{t}^{∗}(γ_{0}, ω) × γ_{t}^{∗}(γ_{0}, ω)⟩^{2}.

The functions A_{n}(t) are non-negative and continuous. There exists a sequence of positive numbers {b_{n}} for which

∑_{n} b_{n}(1 + A_{n}(t)) exp \{\frac{1}{2}ba_{n}^{−1}\} < ∞, t > 0.  (12)

Formulas (11) and (12) imply the relation

lim sup_{c→∞} E ∑_{n} b_{n}⟨f_{n}, γ_{t}^{∗}(γ_{0}, ω) × γ_{t}^{∗}(γ_{0}, ω)⟩ρ(c, τ^{∗}) < ∞.

This formula implies that the set of measures {P_{c}^{∗}, c > 0} for which

\frac{dP_{c}^{∗}}{dP} (ω) = ρ(c, τ^{∗})

is weakly compact, so there exists a sequence c′_{k}, c′_{k} → ∞ for which P_{c′_{k}}^{∗} converges weakly to a measure P^{∗}. The statement (i) is proved.

Introduce the stopping times

τ_{n}^{∗} = inf {t : ∑_{k≤n} a_{k}G_{2}(c_{k}, t) ≥ b},

and set

P_{n}^{∗}(A) = E1_{A}ρ(c_{n}, τ_{n}^{∗}), A ∈ F.

Let Φ(ξ_{1}, …, ξ_{m}) be the same as before. Then

lim_{n→∞} E_{n}^{∗}Φ(ξ_{1}, …, ξ_{m}) = E^{∗}Φ(ξ_{1}, …, ξ_{m});

here E_{n}^{∗} is the expectation with respect to probability P_{n}^{∗}.This formula is a consequence of the relations

τ_{n}^{∗} > τ^{∗}, τ_{n}^{∗} → τ^{∗}

in probability and

E(ρ(c_{n}, τ_{n}^{∗})/F_{τ^{∗}}^{c_{n}}) = ρ(c_{n}, τ∗).

Let w_{i}^{∗}(c, t) be given by formula (8). Then for fixed n the sequence

{w_{i}^{∗}(c_{n}, t), i = 1, 2, … }

represents independent Wiener processes on the probability space {Ω, F, P_{n}^{∗}}. Introduce stopping times

ζ_{i}^{n} = inf{t : |x_{i}(t)| > c_{n} − r}.

Since a_{c}(x, ω) = 0 for |x| < c_{n} − r,we have

w_{i}^{∗}(c_{n}, t ∨ ζ_{i}^{n}) = w_{i}^{∗}(t ∨ ζ_{i}^{n}).

Using Remark 3 we can prove that the relation

lim_{c→∞} lim sup_{n→∞} P_{n}^{∗}{sup_{s≤t} |x_{i}^{∗}(s)| > c} = 0  (13)

is fulfilled for any i and t > 0. Let Φ, h_{1}, …, h_{m} be the same as before . Set

ξ_{i}^{∗} = ∫_{0}^{t} h_{i}(s)dw_{i}^{∗}(s), ξ_{i}^{n} = ∫_{0}^{t} h_{i}(s)dw_{i}^{∗}(c_{n}, s), i = 1, ….

Note that

E_{n}^{∗}Φ(ξ_{1}^{n}, …, ξ_{m}^{n}) = EΦ( ∫_{0}^{t} h_{1}(s)dw_{1}(s), …, ∫_{0}^{t} h_{m}(s)dw_{m}(s))  (14)

for all n. Denote the expression in the right hand side of formula (14) by Φ̄. Then

E_{n}^{∗}Φ(ξ_{1}^{∗}, …, ξ_{m}^{∗}) = Φ̄ + E_{n}^{∗}O( ∑_{i≤m} 1_{{sup_{s≤t} |x_{i}^{∗}(s)|>c_{n}−r}}).  (15)

Formulas (13) and (15) imply the relation

E^{∗}Φ(ξ_{1}^{∗}, …, ξ_{m}^{∗}) = lim_{n→∞} E_{n}^{∗}Φ(ξ_{1}^{∗}, …, ξ_{m}^{∗}) = Φ̄.

The statement (ii) is proved.

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(Received 3.07.2000)

Author’s addresses: Institute of Mathematics National Academy of Sciences of Ukraine 3, Tereshchenkivska St., Kyiv 01601, Ukraine Department of Statistics and Probability Michigan State University A 415 Wells Hall, East Lansing, MI 48824, U.S.A. E-mail: skorokhod@stt.msu.edu