STROOCK-VARADHAN THEOREM FOR FLOWS GENERATED BY STOCHASTIC DIFFERENTIAL EQUATIONS WITH INTERACTION

A. Yu. Pilipenko

We prove a theorem that characterizes the support of a flow generated by a system of stochastic differential equations with interaction.

Consider the system of stochastic differential equations with interaction

\begin{cases}dx(u, t) & = ∫_{ℝ^{d}} a(x(u, t), x(v, t))μ(dv)dt+ \\ & + ∑_{k=1}^{m} ∫_{ℝ^{d}} b_{k}(x(u, t), x(v, t))μ_{k}(dv)) ∘ dw^{k}(t), t ∈ [0, T] \\ x(u, 0) & = φ(u), u ∈ ℝ^{d}, \end{cases}

(1)

where a : ℝ^{d} × ℝ^{d} → ℝ^{d}, b_{k} : ℝ^{d} → ℝ^{d}, φ : ℝ^{d} → ℝ^{d}, μ, μ_{k} are probability measures on ℬ(ℝ^{d}) and w^{1}(t), …, w^{m}(t) are independent one-dimensional Wiener processes.

Under some natural conditions on the smoothness and growth of the coefficients a and b_{k} of the equation and the function φ [1], there exists a unique solution of Eq.(1). In this case, a certain modification of x(u, t) has derivatives with respect to u, which are continuous in (u, t) almost surely. Thus, the flow {x(u, t), u ∈ ℝ^{d}}_{t∈[0,T]} can be regarded as a random element in the space C([0, T] → C^{n}(ℝ^{d}; ℝ^{d})), , where the space C^{n}(ℝ^{d}, ℝ^{d}) of functions from ℝ^{d} into ℝ^{d} that are n times continuously differentiable is equipped with the topology of uniform convergence of derivatives on compacta.

As proved by Stroock and Varadhan (see, e.g., [2]), the support of a diffusion process x(t), t ∈ [0, T], that satisfies the equation

\begin{cases}dx(t) & = a(x(t))dt + ∑_{k=1}^{m} b_{k}(x(t)) ∘ dw^{k}(t), t ∈ [0, T] \\ x(0) & = x_{0} \end{cases}

coincides with the closure of the set of functions {x^{ψ} | ψ = (ψ_{1}, …, ψ_{m}) ∈

C^{2}([0, T]; ℝ^{m})} in the space C([0, T], ℝ^{d}); here, x^{ψ} is defined by the ordinary differential equation

\begin{cases}\frac{dx^{ψ}(t)}{dt} & = a(x^{ψ}(t)) + ∑_{k=1}^{m} b_{k}(x^{ψ}(t)), t ∈ [0, T] \\ x^{ψ}(0) & = x_{0} \end{cases}

An analogous statement on a flow of diffeomorphisms generated by a stochastic differential equation was proved in [3]. The main result of the present paper is the proof of an analog of the Stroock-Varadhan theorem for flows generated by the stochastic equation with interaction (1).

We introduce the following notation: ∥f∥_{T} := sup_{t∈[0,T]} ∥f(t)∥, where the function f acts from [0, T] into the Euclidean space ℝ^{n};

∥g∥_{U} := sup_{u∈U} ∥g(u)∥, for g : U ⊂ ℝ^{d} → ℝ^{d};

∥h∥_{U,T} := sup_{u∈U} sup_{t∈[0,T]} ∥h(u, t)∥, for h : U × [0, T] → ℝ^{d}.

In what follows, as a rule, we omit the indices T and U. For example, ∥w∥ is understood as ∥w∥_{T} = sup_{t} ∥(w(t), …, w_{m}(t))∥_{ℝ^{m}}.

Let ψ = (ψ_{1}, …, ψ_{m}) : [0, T] → ℝ^{m} be a continuously differentiable function.

We denote by x^{ψ} a solution of the following system of differential equations:

\begin{cases}\frac{∂x^{ψ}(u,t)}{∂t} & = ∫_{ℝ^{d}} a(x^{ψ}(u, t), x^{ψ}(v, t))μ(dv)+ \\ & + ∑_{k=1}^{m} ∫_{ℝ^{d}} b_{k}(x^{ψ}(u, t), x^{ψ}(v, t))μ_{k}(dv))(ψ^{k})′(t), t ∈ [0, T] \\ x^{ψ}(u, 0) & = φ(u). \end{cases}

(2)

Theorem 1. Suppose that the following conditions are satisfied:

(i) φ ∈ C^{n}(ℝ^{d}, ℝ^{d});

(ii) the functions a and b_{k}, k = 1, …, m are continuously differentiable with respect to the first variable (n + 2) and (n + 4) times, respectively, and are bounded together with their derivatives;

(iii) the functions a and b_{k} and all their derivatives satisfy the global Lipschitz condition with respect to the second variable, i.e.,

∃L > 0 ∀u, v_{1}, v_{2} ∈ ℝ^{d} ∀l = 0, …, n + 2, p = 0, …, n + 4 :

∥\frac{∂^{l}a(u, v_{1})}{∂u^{l}} − \frac{∂^{l}a(u, v_{2})}{∂u^{l}}∥ ≤ L∥v_{1} − v_{2}∥,

∥\frac{∂^{p}b_{k}(u, v_{1})}{∂u^{p}} − \frac{∂^{p}b_{k}(u, v_{2})}{∂u^{p}}∥ ≤ L∥v_{1} − v_{2}∥;

(iv) μ and μ_{k} are probability measures on ℝ^{d}.

Then, for any function ψ ∈ C^{2}([0, T], ℝ^{m}), a bounded set U ⊂ ℝ^{d} and θ > 0 one has

lim_{δ→0} P (sup_{u∈U} sup_{t∈[0,T]} sup_{0≤l≤n} ∥\frac{∂^{l}x(u, t)}{∂u^{l}} − \frac{∂^{l}x^{ψ}(u, t)}{∂u^{l}}∥ ≥ θ / ∥w − ψ∥ < δ) = 0.

(3)

The main result of the present paper is the theorem below, which is a direct consequence of relation (3).

Theorem 2. The support of the distribution x(u, t), u ∈ ℝ^{d}, t ∈ [0, T] contains the closure of the set {x^{ψ}|ψ ∈ C^{2}([0, T]; ℝ^{m})} in the space C([0, T] → C^{n}(ℝ^{d}; ℝ^{d})).

Remark 1. The author knows the proof of the inverse inclusion. The proof is based, as in [2, 3], on the approximation theorem and will be presented in a separate paper.

Proof of Theorem 1. In [1], it was proved that if the conditions of Theorem 1 are satisfied, then the process x(u, t) has an n-times continuously differentiable modification. In what follows, we consider exactly this modification. It is also known that the derivatives \frac{∂^{k}x(u,t)}{∂u^{k}} satisfy the equations obtained by formally differentiating (1) k times.

First, consider the case n = 0. Assume that ψ = 0. The case of arbitrary ψ can be proved with the use of the Girsanov theorem by analogy with the theorem on the support of diffusion processes (Theorem 8.2 in [2, Chap. 6]). We restrict ourselves to the case where U ⊂ ℝ^{d} is a closed ball.

Let us verify that, for any ε > 0 we have

P (∥∫_{0}^{·} ∫ b_{k}(x(u, t), x(v, t))μ_{k}(dv) ∘ dw^{k}(t)∥_{U,T} ≥ ε/∥w∥ < δ) → 0, δ → 0+.

(4)

It is known [4], that, for any p > d , the Sobolev space W_{p}^{1}(U) is continuously imbedded into C(U). Hence, to prove (4), it suffices to verify that

E (sup_{t} ∫_{U} (∥ ∫_{0}^{t} ∫ b_{k}(x(u, t), x(v, t))μ_{k}(dv) ∘ dw^{k}(s)∥^{p}+  (5)

∥ ∫_{0}^{t} ∫ (b_{k})′_{1}(x(u, s), x(v, s))μ_{k}(dv)\frac{∂x(u, s)}{∂u} ∘ dw^{k}(s)∥^{p}) du/∥w∥ < δ) → 0, δ → 0+.

Estimate (5) follows from the relations

sup_{u∈U} E (∥∫_{0}^{·} ∫ b_{k}(x(u, t), x(v, t))μ_{k}(dv) ∘ dw^{k}(s)∥_{T}^{p} /∥w∥ < δ) → 0, δ → 0+,

(6)

sup_{u∈U} E (∥∫_{0}^{·} ∫ (b_{k})′_{1}(x(u, s), x(v, s))μ_{k}(dv)\frac{∂x(u, s)}{∂u} ∘ dw^{k}(s)∥_{T}^{p} /∥w∥ < δ) → 0, δ → 0+.

(7)

We need the following definition:

Definition. A semimartingale b(t), t ∈ [0, T],

db(t) = b_{0}(t)dt + b_{k}(t)dw^{k}(t)

belongs to the class B_{0}, if there exists a constant β, such that, for any k ∈ {0, …, m}, t ∈ [0, T] we have

|b_{k}(t)| ≤ β_{0} a.s.

If each diffusion characteristic b_{k}(t), 1 ≤ k ≤, is a semimartingale from B_{0}, then we say that b(t) belongs to the class B_{1}.

Similarly, if

db_{k}(t) = b_{k,0}(t)dt + b_{k,j}(t)dw^{j}(t),

where b_{k,j}(t) belongs to B_{0}, 1 ≤ k, j ≤ m, then b(t), t ∈ [0, T] belongs to the class B_{2}, and so on.

Lemma 1. Suppose that a semimartingale b(t), t ∈ [0, T] belongs to the class B_{3}. Then there exist positive constants K_{1}, K_{2} = K_{2}(b), K_{3} = K_{3}(B), and a positive non-decreasing function M = M(δ) such that M(δ) → 0 as δ → 0+ and

∀ε > 0 ∃δ_{0} > 0 ∀δ ∈ (0, δ_{0}) ∀M > M(δ)∀k ∈ {1, …, m} :

P (∥ ∫_{0}^{·} b(s) ∘ dw^{k}(s)∥_{T} ≥ ε/∥w∥ < δ) ≤

≤ K_{1} exp (−\frac{min \{\frac{ε^{2}}{K_{2}}; K_{1}M^{2}; \frac{ε^{2}}{K_{3}M^{2}} − 1\}}{δ^{2}}) .  (8)

Remark 2. It follows from the Stroock–Varadhan theorem that the probability in (8) tends to zero as δ → 0 + . However, we need more exact asymptotics.

Proof of Lemma 1. We apply the formula of integration by parts to the stochastic integral

∫_{0}^{t} b(s) ∘ dw^{k}(s) = b(t)w^{k}(t) − ∫_{0}^{t} w^{k}(s)b̃_{0}(s)ds−

− ∫_{0}^{t} b_{j}(s)w^{k}(s) ∘ dw^{j}(s) = I_{1}(t) + I_{2}(t) + I_{3}(t).  (9)

Here,

db(t) = b_{0}(t)dt + b_{j}(t)dw^{j}(t) = b̃_{0}(t)dt + b_{j}(t) ∘ dw^{j}(t),

and the process b̃_{0}(t), t ∈ [0, T] is obtained by using the Ito formula.

In the proof, we use many constants. To avoid the use of numerous different indices, we write a dot instead of them. For example, K_{·} and K_{·}(b), are understood, respectively, as a universal constant (possibly, dependent on d or T ) and a constant dependent only on the properties of the semimartingale b. Note that

∥I_{1}∥ ≤ K_{·}(b)δ and ∥I_{2}∥ ≤ K_{·}(b)δ  (10)

on the set {∥w∥ < δ}.

Denote ξ^{kj}(t) := ∫_{0}^{t} w^{k}(s) ∘ dw^{j}(s).

Then

I_{3}(t) = −b_{j}(t)ξ^{kj}(t) + ∫_{0}^{t} ξ^{kj}(s)b_{j,0}(s)ds + ∫_{0}^{t} ξ^{kj}(s)b_{j,l}(s) ∘ dw^{l}(s) =

= ℒ_{1}(t) + ℒ_{2}(t) + ℒ_{3}(t).  (11)

In the integral ℒ_{3} we pass from the Stratonovich integral to the Ito integral and apply the Ito formula once again. As a result, we get

ℒ_{3}(t) = α(ξ^{kj}, b)(t) − ∫_{0}^{t} ξ^{kj}(s)w^{l}(s)b_{j,l,i}(s)dw^{i}(s)−

− ∫_{0}^{t} b_{j,l}(s)w^{l}(s)dξ^{kj}(s),  (12)

where ∥α(ξ^{kj}, b)(·)∥_{T} ≤ K_{·}(b)∥ξ^{kj}∥_{T} .

Let M > 0 be a certain number, let β(t), t ∈ [0, T] be a nonanticipating process, and let |β(t)| ≤ B a.s., where B = const > 0.

Then

P (∥ ∫_{0}^{·} ξ^{kj}(s)w^{l}(s)β(s)dw^{i}(s) ≥ ε/∥w∥ < δ) ≤
≤ P (∥ξ^{kj}∥ ≥ M/∥w∥ < δ) +
+P (∥ξ^{kj}∥ < M, ∥ ∫_{0}^{·} ξ^{kj}(s)w^{l}(s)β(s)dw^{i}(s)∥ ≥ ε/∥w∥ < δ) . (13)

Lemma 2. There exist positive constants c_{1} and c_{2} such that, for any M > 0, δ > 0 and M > δ^{2} , the following inequality is true:

P (∥ξ^{kj}∥ ≥ M/∥w∥ < δ) ≤ \frac{c_{1}δ}{M}exp \{−\frac{c_{2}M^{2}}{δ^{2}}\} .  (14)

Proof. Assume that k ≠ j and

η^{kj}(t) = \frac{1}{2} ∫_{0}^{t}[w^{k}(s)dw^{j}(s) − w^{j}(s)dw^{k}(s)],

a(t) = \frac{1}{4} ∫_{0}^{t}[(w^{k})^{2}(s) + (w^{j})^{2}(s)]ds.

It is proved in [2, Chap. 6, Sec. 6], that B(t) ≡ η^{kj}(a^{−1}(t)) is a Brownian motion independent of {(w^{k})^{2}(t) + (w^{j})^{2}(t)}. Therefore, by analogy with Lemma 8.2 in [2], we obtain

P {∥η^{kj}∥_{T} ≥ M/∥w∥ < δ} = P {∥B(a(t))∥_{T} ≥ M/∥w∥ < δ} ≤

≤ P { max_{0≤s≤T δ^{2}/4} |B(s)| > M} <

< 4P \{B(T δ^{2}/4) > M\} = \frac{4}{\sqrt{2π}} ∫ ∞_{\frac{2M}{\sqrt{T} δ}} e^{−\frac{x^{2}}{2}} dx ≤ \frac{2\sqrt{T} δ}{M} exp \{−\frac{2M^{2}}{T δ^{2}}\} .

(15)

Inequality (14) follows directly from relation (15) in view of the fact that

ξ^{kj}(t) = −η^{kj}(t) + \frac{1}{2}w^{k}(t)w^{j}(t).

If k = j, then the statement of Lemma 2 is obvious because ξ^{kk}(t) = \frac{1}{2}(w^{k})^{2}(t).

Lemma 2 is proved.

Lemma 3 [2, Lemma 8.1 in Chap. 6]. There exist constants c_{3} and c_{4}, such that

P (∥w∥ < δ) ∼ c_{3} exp (\frac{−c_{4}}{δ^{2}}) , δ → +0.

To estimate the right-hand side of (13), we use the following known fact: If M(t) is a square-integrable martingale with characteristic ⟨M(t)⟩, and continuous trajectories and M(0) = 0, then there exists a Brownian motion W (possibly, on an extension of the probability space) such that M(t) = W (⟨M⟩(t)). Hence, the second term on the right-hand side of (13) does not exceed the following value:

P { sup_{t∈[0,T]} ∥W (tB^{2}δ^{2}M^{2})∥ ≥ ε}/P {∥w∥ < δ} ≤

≤ P \{ sup_{s∈[0,T]} ∥W (s)∥ ≥ \frac{ε}{BδM} \}/ P \{∥w∥ < δ\} ≤

≤ K_{·} exp \{\frac{(\frac{ε^{2}}{B^{2}M^{2}} − 1) K_{·}}{δ^{2}} \} .

The last term in (12) can be examined by analogy with regard for the fact that

∫_{0}^{t} b_{j,l}(s)w^{l}(s)dξ^{kj}(s) = ∫_{0}^{t} b_{j,l}(s)w^{l}(s)w^{k}(s)dw^{j}(s)+

+\frac{δ_{k,j}}{2} ∫_{0}^{t} b_{j,l}w^{l}(s)ds.

The statement of Lemma 1 now follows from estimates (9)–(13).

We continue the proof of Theorem 1. Let us verify the convergence of (6). Let u ∈ U. Note that the random process ∫ b_{k}(x(u, t), x(v, t))μ_{k}(dv), t ∈ [0, T], satisfies the conditions of Lemma 1. Set ξ(u) = ∥ ∫_{0}^{·} ∫ b_{k}(x(u, s), x(v, s))μ_{k}(dv)∘ dw^{k}(t)∥. Hence,

E (ξ(u)^{p}/∥w∥ < δ) ≤
≤ p ∫_{0}^{1} y^{p−1}P (ξ(u) > y/∥w∥ < δ)dy+

+\frac{E (ξ(u)^{p}1I_{\{ξ(u)≥1\}}1I_{\{∥w∥<δ\}})}{P (∥w∥ < δ)} = I_{1}(δ) + I_{2}(δ).  (16)

It is easy to see [see estimate (8)] that, for any y > 0, we have

sup_{u∈U} P (∥∫_{0}^{·} ∫ b_{k}(x(u, s), x(v, s))μ_{k}(dv) ∘ dw^{k}(t)∥ > y/∥w∥ < δ) → 0, δ → 0+.

Hence, by virtue of the Lebesgue dominated-convergence theorem, we have convergence to zero in (16) uniform with respect to u ∈ U.

Note that sup_{u} Eξ(u)^{p} < ∞ for each p > 0.

Therefore, due to Lemmas 1 and 3:

I_{2}(δ) ≤ (Eξ(u)^{α})^{1/α} [P (ξ(u) > 1, ∥w∥ < δ)]^{1/β} /P (∥w∥ < δ) =

= c_{·} (P (ξ(u) > 1/∥w∥ < δ))^{1/β} (P (∥w∥ < delta))^{−1/α} ≤

≤ exp \{\frac{−\frac{1}{β} (min\{\frac{1}{K_{2}}, K_{1}M, \frac{1}{K_{3}M^{2}} − 1\}) + \frac{c_{·}}{α}}{δ^{2}} \} ,  (16.5)

where 1/α + 1/β = 1.

Choose M and α so that the expression in the numerator of (16.5) be negative and let δ → 0+. Then I_{2}(δ) → 0.

Note that the integrand in (7) does not satisfy the conditions of Lemma 1 because, generally speaking, \frac{∂x(u,t)}{∂u} is unbounded. Therefore, we need some additional arguments.

Assume that c > 0 and f_{c} : ℝ^{d} → ℝ^{d} is a finite, nonnegative, infinitely differentiable function such that f_{c}(x) = x for |x| ≤ c.

Denote by y(u, t) a solution of the equation

\begin{cases}dy(u, t) & = ∫_{ℝ^{d}} a′_{1}(x(u, t), x(v, t))μ(dv)f_{c}(y_{c}(u, t))dt+ \\ & ∑_{k=1}^{m} ∫ (b_{k})′_{1}(x(u, s), x(v, s))μ_{k}(dv)f_{c}(y_{c}(u, t)) ∘ dw^{k}(t) \\ y_{c}(u, 0) & = ∇φ(u). \end{cases}

(17)

Note that

P ({ω : ∥y_{c}(u, t, ω)∥ ≤ c, u ∈ U, t ∈ [0, T]}△

△\{ω : ∥\frac{∂x(u, t, ω)}{∂u}∥ ≤ c, u ∈ U, t ∈ [0, T]\}) = 0

and, on the set \{sup_{u∈U,t∈[0,T]} ∥\frac{∂x(u,t)}{∂u}∥ ≤ c\} the following equality holds almost surely:

y_{c}(u, t) = \frac{∂x(u,t)}{∂u} for all t ∈ [0, T], u ∈ U. Assume that ω and u are such that

∥ ∑_{k=1}^{m} ∫_{0}^{·} ∫ (b_{k})′_{1}(x(u, s), x(v, s))μ_{k}(dv)\frac{∂x(u, t)}{∂u} ∘ dw^{k}(t)∥_{T} ≤ ε.

Then, applying the Gronwall-Bellman lemma to the equations for the derivative

\begin{cases}d\frac{∂x(u,t)}{∂u} & = ∫_{ℝ^{d}} a′_{1}(x(u, t), x(v, t))μ(dv)\frac{∂x(u,t)}{∂u}dt+ \\ & ∑_{k=1}^{m} ∫ (b_{k})′_{1}(x(u, s), x(v, s))μ_{k}(dv)\frac{∂x(u,t)}{∂u} ∘ dw^{k}(t) \\ \frac{∂x(u,t)}{∂u} & = ∇φ(u). \end{cases}

we obtain

∥\frac{∂x(u, t)}{∂u}∥ ≤ (∥∇φ(u)∥ + ε) exp(∥a′_{1}∥_{∞}T).  (18)

Let ε > 0 be fixed. We choose c > 0, so that the right-hand side of (18) does not exceed cfor all u ∈ U. Then y_{c}(u, t) = \frac{∂x(u,t)}{∂u}, u ∈ U, t ∈ [0, T] for almost all ω from the set

{∥ ∑_{k=1}^{m} ∫_{0}^{·} ∫ (b_{k})′_{1}(x(u, s), x(v, s))μ_{k}(dv)f_{c}(y_{c}(u, t)) ∘ dω^{k}(t)∥_{T} ≥ ε}. De-

note

ξ_{t}(u) := ∑_{k=1}^{m} ∫_{0}^{t} ∫ (b_{k})′_{1}(x(u, s), x(v, s))μ_{k}(dv)\frac{∂x(u, s)}{∂u} ∘ dω^{k}(s).

Then

E(∥ξ_{·}(u)∥_{T}^{p} /∥w∥ < δ) = E(∥ξ_{·}(u)∥_{T}^{p} 1I_{{∥ξ_{·}(u)∥_{T} <ε}}/∥w∥ < δ)+

+E(∥ξ_{·}(u)∥_{T}^{p} 1I_{{∥ξ_{·}(u)∥_{T} ≥ε}}/∥w∥ < δ) = I_{1}(δ) + I_{2}(δ).

The proof of the convergence of I_{1}(δ) to zero as δ → 0 is similar to the proof of (6) because

I_{1}(δ) ≤ E(∥ ∑_{k=1}^{m} ∫_{0}^{·} ∫ b_{k}(x(u, s), x(v, s))μ_{k}(dv)f_{c}(y_{c}(u, s))∘dw^{k}(s)∥_{T}^{p} /∥w∥ < δ).

To estimate I_{2}(δ), we use the Holder inequality

I_{2}(δ) ≤ \frac{(E∥ξ_{·}(u)∥_{T}^{pα})^{1/α}(P (∥w∥ < δ, ∥ξ_{·}(u)∥_{T} ≥ ε))^{1/β}}{P (∥w∥ < δ)} ,  (19)

where \frac{1}{α} + \frac{1}{β} = 1.

It is easy to verify that K(U, α, p) = (E∥ξ_{·}(u)∥_{T}^{pα})^{1/α} < ∞. Hence, the expression in (19) does not exceed the following value (see Lemmas 1 and 3):

K(U, α, p) sup_{u∈U} \frac{[P (∥ ∑_{k=1}^{m} ∫_{0}^{·} ∫ b_{k}(x(u, s), x(v, s))μ_{k}(dv)f_{c}(y_{c}(u, s)) ∘ dω^{k}(s)∥_{T} ≥ ε/∥w∥ < δ)]}{[P (∥w∥ < δ)]^{1/α}} ≤

≤ K(U, α, p) exp[\{−\frac{1}{β} (\frac{θ^{2}}{K(b_{k}, U, ε, f_{c})}) ∧ K_{·}M^{2}∧

∧ (\frac{θ^{2}}{K(b_{k}, U, ε, f_{c})M^{2}} − 1) + \frac{c_{2}}{α}\} \frac{1}{δ^{2}}].  (20)

We choose M and then α so that the quantity in the numerator in

(20) is negative. Then I_{2}(δ) → 0 as δ → 0 + . Thus, we have proved relation (7) and, consequently, relation (4).

We now pass to the proof of (3) for ψ = 0. Let Ũ ⊂ ℝ^{d} be a closed

ball and let u ∈ Ũ. Denote o(Ũ) := ∥ ∫_{0}^{·} b_{k}(x(u, s)) ∘ dw^{k}(s)∥_{Ũ,T} . Then

sup_{s∈[0,T]} ∥x(u, s) − x^{0}(u, s)∥ ≤

≤ ∫_{0}^{t} ∫Ũ ∥a(x(u, s), x(v, s)) − a(x^{0}(u, s), x^{0}(v, s))∥μ(dv)ds+

+2 sup_{x,y} ∥a(x, y)∥μ(ℝ^{d} \ Ũ) + o(Ũ) ≤

≤ K_{·}(a)(∫_{0}^{t} sup_{z∈[0,s]} ∥x(u, s) − x^{0}(u, s)∥ds+

+ ∫_{0}^{t} ∫Ũ sup_{z∈[0,s]} ∥x(v, s) − x^{0}(v, s)∥μ(dv)ds + μ(ℝ^{d} \ Ũ)) + o(Ũ). (21)

Integrating (21) with respect to the parameter u ∈ Ũ and replacing u by v, we obtain

∫Ũ sup_{s∈[0,t]} ∥x(v, s) − x^{0}(v, s)∥μ(dv) ≤

≤ K_{·}(a) (∫_{0}^{t} ∫Ũ sup_{z∈[0,s]} ∥x(v, s) − x^{0}(v, s)∥μ(dv)ds + μ(ℝ^{d} \ Ũ)) + o(Ũ).

(22)

Applying the Gronwall-Bellman lemma to (22), substituting the expression obtained into (21), and applying the Gronwall–Bellman lemma once again, we get

sup_{s∈[0,t]} ∥x(u, s) − x^{0}(u, s)∥ ≤ K_{·}(a, T)(μ(ℝ^{d} \ Ũ) + o(Ũ)), t ∈ [0, T].

Let θ > 0 and ε > 0 be fixed. We choose a set Ũ that contains U so that

K_{·}(a, T)μ(ℝ^{d} \ Ũ) < \frac{θ}{2}.

Then [see (4)]

lim_{δ→0+} P ( sup_{tı[0,T]} sup_{u∈U} ∥x(u, s) − x^{0}(u, s)∥ ≥ θ/∥w∥ < δ) ≤

≤ lim_{δ→0+} P (K_{·}(a, T)o(Ũ) ≥ \frac{θ}{2}/∥w∥ < δ) = 0.

Theorem 1 is proved in the case n = 0. The general case differs from the case n = 1 insignificantly, and, therefore, we restrict ourselves to the question about the support of the random process x(u, t) in the space C([0, T] → C^{1}(ℝ^{d}, ℝ^{d})). We consider Eq. (1) and the equations for the derivative y(u, t) := \frac{∂x(u,t)}{∂u},

\begin{cases}dy(u, t) & = [∫_{ℝ^{d}} a′_{1}(x(u, t), x(v, t))μ(dv)]y(u, t)dt+ \\ & ∑_{k=1}^{m} ∫_{ℝ^{d}}(b_{k})′_{1}(x(u, t), x(v, t))μ_{k}(du)y(u, t) ∘ dω^{k}(t) \\ y(u, 0) & = \frac{∂φ(u)}{∂u}, \end{cases} (23)

as a system of stochastic equations with interaction. Unfortunately, Eqs.

(1) and (23) do not satisfy the conditions of Theorem 1 for n = 0 because the coefficients in the equation for y(u, t) are unbounded. To prove the theorem, we use the localization principle. Let y_{c}(u, t) and f_{c} be the same as in (17).

Let

τ_{c}(U) = inf{t|∃u ∈ U, ∥y(u, t)∥ ≥ c},
τ̃_{c}(U) = inf{t|∃u ∈ U, ∥y_{c}(u, t)∥ ≥ c} −

be the exit times of y(u, t) and y_{c}(u, t), u ∈ U, respectively, from the ball of radius c centered at the origin. Note that τ_{c}(U) = τ_{c}(Ũ) a.s. and y(u, t) = y_{c}(u, t) a.s. for all u ∈ U, t ∈ [0, τ_{c}(U)].

Let θ > 0 be given. As in the beginning of the proof of Theorem 1, we assume that ψ = 0. Denote by y^{0}(u, t) and y_{c}^{0}(u, t) solutions of the following equations:

\frac{∂y^{0}(u, t)}{∂t} = ∫_{ℝ^{d}} a′_{1}(x^{0}(u, t), x^{0}(v, t))μ(dv)y^{0}(u, t)

\frac{∂y_{c}^{0}(u, t)}{∂t} = ∫_{ℝ^{d}} a′_{1}(x^{0}(u, t), x^{0}(v, t))μ(dv)f_{c}(y_{c}^{0}(u, t))

y^{0}(u, 0) = y_{c}(u, 0) = \frac{∂φ(u)}{∂u},

where x^{0} is a solution of (2) for ψ = 0.

We choose c > 0 so that

sup_{u∈U,t∈[0,T]} ∥y^{0}(u, t)∥ < c − θ.

Then y^{0}(u, t) = y_{c}^{0}(u, t) for all u ∈ U and t ∈ [0, T], and

lim_{δ→0} P (∥y − y^{0}∥_{U,T} ≤ θ/∥w∥ < δ) =

= lim_{δ→0} P (∥y − y^{0}∥_{U,T} ≤ θ, τ_{c}(U) > T/∥w∥ < δ) =

= lim_{δ→0} P (∥y_{c} − y_{c}^{0}∥_{U,T} ≤ θ/∥w∥ < δ) = 1,

because relations (1) and (17), regarded as a system of stochastic equations with values in ℝ^{d+d^{2}}, satisfy the conditions of Theorem 1 for n = 0.

Theorem 1 is proved.

References

[1] A.A.Dorogovtsev, P.Kotelenez Smooth stationary solutions of quasilinear stochastic partial differential equations: 1.Finite mass// Probability theory and its applications,–2002(to appear)

[2] Ikeda N., Watanabe S. Stochastic differential equations and diffusion processes, North Holland-Kodansha, Tokyo, 1981.

[3] Kunita H. Stochastic flows and stochastic differential equations. Cambridge studies in advanced mathematics: 24, 1990, 346 p.

[4] O. V. Besov, V. P. Ilin, and S. M. Nikolskii, Integral Representations of Functions and Imbedding Theorems [in Russian], Nauka, Moscow (1996).