The small deviations of many-dimensional diffusion processes and rarefaction by boundaries

TL;DR

This paper develops an expansion algorithm for the sojourn probability of high-dimensional diffusion processes in small domains, which aids in establishing limit theorems by defining a key normalizing coefficient.

Contribution

It introduces a novel expansion algorithm for the sojourn probability in many-dimensional diffusion processes, facilitating new limit theorems.

Findings

Derived the principal term of the expansion as a normalizing coefficient

Established connections between the expansion and limit theorems

Provided a method for analyzing boundary effects in diffusion processes

Abstract

We lead the algorithm of expansion of sojourn probability of many-dimensional diffusion processes in small domain. The principal member of this expansion defines normalizing coefficient for special limit theorems.

Equations

$$P\left(\xi (t)\in ϵD,t\in [0,T]\right)$$

$$P(ϵ,x)=P\left(\xi (t)\in ϵD,0\le t\le T\right),ϵ\to 0,$$

$$d\xi (t)=a(t,\xi (t))dt+\sum _{i=1}^d{b}_i(\xi (t))d{w}_i(t),\xi (0)=x\in ϵD.$$

$$b_i(x),a(t,x):R^d\to R^{d}and{R}_{+}\times R^d\to R^d.$$

$$\frac{\partial u_0^{ϵ}(t,x)}{\partial t}=\frac{1}{2}\sum _{i,j=1}^d{\sigma }_{ij}(x)\frac{{\partial }^2{u}_0^{ϵ}(t,x)}{\partial x_i\partial x_j}+\sum _{i=1}^d{a}_i(T-t,x)\frac{\partial u_0^{ϵ}(t,x)}{\partial x_i},x\in D_{ϵ};$$

$$u(t,x){|}_{t=0}=1;x\in D_{ϵ};u(t,x)=0x\in \partial D_{ϵ},0\le t\le T.$$

$$\sum _{1\le i,j\le d}{\sigma }_{ij}(0)z_i{z}_j\ge \mu |\vec{z}{|}^2.$$

$$H_A=\{u:u\in L_2(D)\cap Au\in L_2(D)\cap u(\partial D)=0\}$$

$$Au=-\lambda u,u(\partial D)=0$$

$$0<{\lambda }_1<{\lambda }_2<\cdots <{\lambda }_s<\cdots .$$

$$f_{11},\dots ,f_{1n_1},\cdots ,f_{s1},\dots ,f_{s{n}_s},\cdots$$

$$e(x,y,\lambda )=\sum _{{\lambda }_j\le \lambda }{f}_j(x)f_j(y).$$

$$\underset{x,y\in D}{\sup }\sqrt{|D_{x,y}^{\alpha }e(x,y,\lambda )|}\le C_{\alpha }{\lambda }^{(n+|\alpha |)/2}$$

$$\underset{(t,z)\in [0,T]\times D,1\le i,j\le d}{\sup }\max \left\{|\frac{\partial a_i(t,z)}{\partial z_j}|,|\frac{\partial b_i(z)}{\partial z_j}|,|\frac{\partial a_i(T-t,z)}{\partial t}|\right\}<\infty$$

$$P(ϵ,zϵ)=\exp \left\{-{\lambda }_1\frac{T}{{ϵ}^2}+{\int }_0^T\mu (t)dt\right\}\sum _{m=1}^{n_1}{c}_{1m}{f}_{1m}(z)\left(1+O(ϵ)\right),atz\in D,$$

$$\mu (t)=\sum _{i,j}\left(\frac{1}{2}{\sigma }_{ij}(0)a_i(t,0)a_j(t,0)-{\delta }_{ij}{a}_i(t,0)a_j(t,0)\right)$$

$$x_i=z_iϵ,u_1^{ϵ}=u_0^{ϵ}\exp \left\{ϵ\sum _{k=1}^d{a}_k(T-t,0)z_k\right\},$$

$$\frac{\partial u_1^{ϵ}(t,z)}{\partial t}=\frac{1}{2{ϵ}^2}\sum _{i,j=1}^d{\sigma }_{ij}(ϵz)\frac{{\partial }^2{u}_1^{ϵ}(t,z)}{\partial z_i\partial z_j}+\frac{1}{ϵ}\sum _{i,j}\left(a_i(T-t,ϵz)-\frac{1}{2}{\sigma }_{ij}(ϵz)a_j(T-t,0)\right)\frac{\partial u_1^{ϵ}(t,z)}{\partial z_i}$$

$$+\sum _{i,j}\left(\frac{1}{2}{\sigma }_{ij}(ϵz)a_i(T-t,0)a_j(T-t,0)-{\delta }_{ij}{a}_i(T-t,0)a_j(T-t,ϵz)-ϵ\frac{\partial a_i(T-t,0)}{\partial t}{z}_i\right)u_1^{ϵ},z\in D;$$

$$u_1^{ϵ}(t,z){|}_{t=0}=\exp \left\{ϵ\sum _{k=1}^d{a}_k(T,0)z_k\right\};z\in D;u_1^{ϵ}(t,z)=0z\in \partial D,0\le t\le T.$$

$$u_1^{ϵ}(t,z)=\sum _{k\ge 0}{v}_k(t,z){ϵ}^k.$$

$$\exp \left\{ϵ\sum _{k=1}^d{a}_k(T,0)z_k\right\}=1+ϵ\sum _{k=1}^d{a}_k(T,0)z_k+\frac{1}{2!}{\left(ϵ\sum _{k=1}^d{a}_k(T,0)z_k\right)}^2+\cdots ,$$

$$v_0(0,z)=1,v_1(0,z)=\sum _{k=1}^d{a}_k(T,0)z_k,v_2(0,z)=\frac{1}{2}{\left(\sum _{k=1}^d{a}_k(T,0)z_k\right)}^2\cdots .$$

$${\sigma }_{ij}(ϵz)={\sigma }_{ij}(0)+ϵ{\sigma }_{ij}^{ϵ}(z),a_i(T-t,ϵz)=a_i(T-t,0)+ϵa_i^{ϵ}(T-t,z),1\le i,j\le d$$

$$\underset{z\in D,ϵ\in [0,1],1\le i,j\le d}{\sup }|{\sigma }_{ij}^{ϵ}(z)|<\infty ,\underset{z\in D,t\in [0,T],ϵ\in [0,1],1\le i\le d}{\sup }|a_i^{ϵ}(T-t,z)|<\infty$$

$$\frac{\partial v_0}{\partial t}=\frac{1}{2{ϵ}^2}\left(\sum _{i,j=1}^d{\sigma }_{ij}(0)\frac{{\partial }^2}{\partial z_i\partial z_j}\right)v_0+\mu (t)v_0$$

$$v_0{|}_{\partial D}=0;v_0(0,z)=1,z\in D.$$

$$\mu (t)=\sum _{i,j}\left(\frac{1}{2}{\sigma }_{ij}(0)a_i(T-t,0)a_j(T-t,0)-{\delta }_{ij}{a}_i(T-t,0)a_j(T-t,0)\right).$$

$$B^{ϵ}(t,z)f=$$

$$\frac{1}{2ϵ}\sum _{i,j=1}^d{\sigma }_{ij}^{ϵ}(z)\frac{{\partial }^{2}f}{\partial z_i\partial z_j}+\frac{1}{ϵ}\sum _{i,j}\left(a_i(T-t,ϵz)-\frac{1}{2}{\sigma }_{ij}(ϵz)a_j(T-t,0)\right)\frac{\partial f}{\partial z_i}+$$

Full text

Introduction.

Let $\xi(t)$ be a random process with measurable phase space $(X,\Sigma(X))$. Consider the measurable connected domain $D\in\Sigma(X)$ and small parameter $\epsilon$. The investigations of asymptotics of sojourn probability (small deviations)

[TABLE]

is jointed with many practice and theoretical problems [1-4]. In the literature, it was researched both rough asymptotics of principal member of (1)(log from it)[5] and exact asymptotics of diffusion processes of (1)[6-8]. In the works [9,10] was proved of algorithms of expansions of exact asymptotics of small deviation for diffusion and piecewise deterministic random processes for one-dimensional case.

The purpose this article is to present the algorithm of expansion of small deviation for many-dimensional diffusion processes and to define all constants of principal member.

In Section 1 our main result is stated and proved. In section 2 we consider the limits theorems about numbers of unabsorbed diffusion particles by boundaries of small domain.

I. The expansion.

We shall investigate of asymptote of following probability

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where $\xi(t)\in R^{d}$ is solution of the following stochastic differential equation

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where functions

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are differentiable.

Set $\sigma_{ij}(x)=\sum\limits_{k}b_{k}^{i}(x)b_{k}^{j}(x)$.

It is known that $P(\epsilon,x)=u_{0}^{\epsilon}(T,x)$. Here $u_{0}^{\epsilon}(t,x)$ is solution of the following parabolic boundary problem at $0\leq t\leq T$

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where $D_{\epsilon}=\epsilon D$. It is assumed that $D$ is a connected bounded domain from $R^{m}$; the boundary $\partial Q$ is the Lyapunov surface  $C^{(1,\lambda)}$ and $0\in D$. We interest of the asymptotic expansion $\epsilon\to 0$ of solution this problem $u_{0}^{\epsilon}(t,x)$ at $\epsilon\to 0$.

We define the differential operator $A:\frac{1}{2}\sum\limits_{1\leq i,j\leq d}\sigma_{ij}(0)\frac{\partial^{2}}{\partial x_{i}\partial x_{j}}.$ Let $\sigma$ be a matrix with the following property

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Here  $\mu$,  there is a fixed positive number, and $\vec{z}=(z_{1},\cdots,z_{d})$  is an arbitrary real vector.

This operator acts in the following space

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with inner product $(u,v)_{A}=(Au,v)$. Here $(,)$ is inner product in $L_{2}(Q)$. The operator  $A$  is a positive operator[11]. It is known that the following eigenvalue problem

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has infinite set of real eigenvalues $\lambda_{i}\to\infty$ and

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The corresponding eigenfunctions

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form the complete system of functions both in $H_{A}$ and $L_{2}^{0}(Q):=\{u:u\in L_{2}(Q)\cap u(\partial Q)=0\}$. Here the number $n_{k}$ is equal to multiplicity of eigenvalue $\lambda_{k}$.

It is often convenient to present the system of eigenfunctions by one index: $\{f_{n}(z)\}$. The corresponding system of eigenvalues $\{\lambda_{n}\}$ will be with recurrences. We shall use it too.

We introduce the spectral function

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We shall need in the following theorem from the monograph [12].

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Here $\alpha$ is multi-index.

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then the following relation takes place at $\epsilon\to 0$

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where

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and $c_{1m}=\int\limits_{D}f_{1m}(z)dz$.

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Now we obtain the following parabolic problem for function $u_{1}^{\epsilon}$

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We will construct the asymptotic expansion of solution for this initial - boundary problem in the following form

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Note that the famous expansion

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defines the initial conditions for $v_{k},\quad k\geq 0$:

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Using the first fragment of Taylor series in zero point under conditions of theorem we can obtain the following representations

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where

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Now, after substitution of (5),(6) to (4) we conclude that the $v_{0}$ satisfies the problem

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Here

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Further, let us denote by $B_{\epsilon}(t,z)$ the operator $C^{2}(D)\to C(D)$, for $f\in C^{2}(D)$ it’s defined as follows:

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Now, formally the functions $v_{k},k\geq 1$ are defined by the following recurrence system problems

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We shall solve the problems of (7),(8) by method of separation of variables. According to this method the solutions are defined in the form

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For definition of principal number it suffices to construct of the $v_{0}$. If we substitute (9) at $k=0$ to (7) then we obtain

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Set $c_{0,n}=\int_{D}f_{n}(z)dz$ (coefficients of expansion of indicator of set $D$). The initial condition of $v_{0}$ has the following stating

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By definition of system of functions $\{f_{n}(z)\}$, now we have the system of ordinary differential equations

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From the latter one we have

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Set

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We have the following relations for eigenvalues $\lambda_{l}$

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,

Applying Cauchy-Bunyakovskii inequality, Theorem 1 and the latter one, we get

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Here $K_{0}<\infty$.

Reasoning similarly we convince ourselves that for other parts of $B^{\epsilon}(t,z)v_{0}$ the following estimations take place

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where $\max\{K_{0,1},K_{0,2}\}<\infty.$

Now let us estimate the coefficients $\beta^{\epsilon}_{n}(t)$ of expansion of $B^{\epsilon}(t,z)v_{0}$ by system $\{f_{n}\}_{n\geq 1}$. Applying (10)-(12) and Cauchy-Bunyakovskii inequality, we get

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The latter one now gives

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where

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Finally, let us estimate the difference $r^{\epsilon}(t,z)=u^{\epsilon}_{1}(t,z)-v_{0}(t,z)$. By definition, $r^{\epsilon}(t,z)$ is solution of the following problem

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It is clear that $r^{\epsilon}(0,z)$ we can present as $\epsilon r_{1}^{\epsilon}(0,z)$, where $r_{1}^{\epsilon}(0,z)$ is uniform bounded function of variables $\epsilon\in[0,1]$ and $z\in D$. So, the coefficients of expansion this function by system $\{f_{n}(z)\}$ have the following forms

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Now we have the solution of (14) in the following form

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Applying latter one ,(13),(15), Theorem 1 and Cauchy-Bunyakovskii inequality we get at $t>0$

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The proof of theorem is completed.

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Here $\mu^{\epsilon}_{k-1,n}(t)=\int\limits_{D}f_{n}(z)B^{\epsilon}(t,z)v_{k-1}(t,z)dz.$

II. The rarefaction of set of diffusion processes by boundaries of small domains.

The following problem was investigated in works[13,14]. Let a set identical diffusion random processes start at the initial time from the different points of domain $D$. These processes are diffusion processes with absorbtion on the boundary $\partial D$. We are interested in distribution of the number yet absorbed at the moment $T$. The initial number and initial position of diffusion processes are defined either a random Poisson measure[14] or deterministic measure [13]. The proved limits theorems described the situation when $T\to\infty$ and initial number of diffusion processes depended on $T$ and it increased at the rise of $T$. The role of normalizing function played principal member of asymptote of solution of according parabolic problem at $T\to\infty$.

Henceforth we shall assume that considered diffusion processes satisfy of the SDE (2) with different initial points.

Now we consider the situation when initial number of absorbing diffusion processes in small domain $\epsilon D$ depends on $\epsilon\to 0$ and it increase under the condition of decrease of $\epsilon$. It is not hard to show, that now normalizing function is the principal member of parabolic problem (3) at $\epsilon\to 0$.

The proofs of stated below theorems repeat the proofs of according theorems from [13,14] almost word for word.

We will denote by  $\eta(\epsilon,T)$  the number of remaining processes in the region  $\epsilon D$  at the moment  $T$.

We will also assume that $\sigma$-additive measure $\nu$ is given on the $\Sigma_{\nu}$- algebra sets from $D,\quad\nu(D)<\infty.$ All eigenfunctions $f_{ij}:D\to R^{1}$ are $(\Sigma_{\nu},\Sigma_{Y})$ measurable. Here $\Sigma_{Y}$ is system of Borel sets from $R^{1}$. Let $\Rightarrow$ denote the weak convergence of random values or measures.

At the beginning we assume that initial number and position of diffusion processes are defined by deterministic measure $N(\epsilon B,\epsilon),B\in D$. Thus, $N(\epsilon B,\epsilon)$ is equal to number of starting points in the set $\epsilon B$.

Let us denote by  $\nu_{\epsilon}(\cdot)$ the measure

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where $B\in\Sigma_{\nu}$ .

By definition of measure  $\nu_{\epsilon}(\cdot)$, we have

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Then $\eta(\epsilon,T)\Rightarrow\eta(T)$ if $\epsilon\to 0$ where $\eta(T)$ has Poisson distribution function with parameter

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where $F(z)=\sum\limits_{i=1}^{n_{1}}f_{1i}(z)c_{1i},\quad c_{1i}=\int\limits_{Q}f_{1i}(z)dz$

and $\mu(t)$ is the function from Theorem 2.

Now we consider the case when the initial number and positions of processes are defined by the random Poisson measure $\mu(\cdot,\epsilon)$ in $\epsilon D$:

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where $m(\epsilon~{}\cdot,\epsilon)$ is finitely additive positive measure on  $\epsilon D$ for fixed $\epsilon$.

We assign

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Then $\eta(\epsilon,T)\Rightarrow\eta(T)$ if $\epsilon\to 0$ where $\eta(T)$ has the Poisson distribution function with the parameter $a(T)$ from Theorem 3.

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