---
address:
- 'Universita Degli Studi Di Salerno, Via Ponte Don Melillo, 84084 Fisciano (SA) Italia'
- 'Institute of Mathematics,National Academy of Science of Ukraine, Tereshchenkivska 3, 01601, Kiev, Ukraine'
author:
- 'Aniello Fedullo and Vitalii A. Gasanenko'
title: ' Limit theorems for number of diffusion processes which did not absorb by boundaries. '
---

amstex

We have random number of independent diffusion processes with absorption on boundaries in some region at initial time $t=0$. The initial numbers and positions of processes in region is defined by Poisson random measure. It is required to estimate of number of the unabsorbed processes for the fixed time  $\tau>0$. The Poisson random measure depends on $\tau$ and $\tau\to\infty$.

Consider the set of independent random diffusion processes    $\xi_{k}(t)
,\quad k=\overline{1,N},$

$t\geq 0,~~\xi_{k}(0)=x_{k},~~
x_{k}\in Q\subset R^{d}$. We wish to investigate of distribution of the number of the processes  $\xi_{k}(t)$  which was into $Q$ for all moments of time $t\leq \tau$.

Let domain   $ Q\subset R^{d}$   be open connected region and it is limited by smooth surface $\partial Q$. All processes  $\xi_{k}(t)$  are diffusion processes with absorption on the boundary  $\partial Q$. These processes are solutions of the following stochastic differential equations in $Q$

$$d\xi(t)=a(t,\xi(t))dt + \sum\limits_{i=1}^{d}b_{i}(t,\xi (t))dw^{(k)}_{i}(t)
\quad \xi(t)\in R^{d}\eqno(1)$$

$$b_{i}(t,x),~ a(t,x): R_{+}\times R^{d}\to R^{d}.$$

with an initial condition:  $\xi(0)=x_{k}\in D.$

Here the  $W^{(k)}(t)=(w_{i}^{(k)}(t),\quad 1\leq i\leq d),\quad 1\leq k\leq N$  are independent in totality $d$- dimensional Wiener processes.

Thus, these processes have the identical diffusion matrices and shift vectors , but they have different initial states.

Let $Q$ is bounded and boundary $\partial Q$ is Lyapunov surface  $C^{(1,\lambda)}$.

The initial number and positions of processes are defined by the random Poisson measure $\mu(\cdot,\tau)$ in $Q$: $$P(\mu(A,\tau)=k)=\frac{m^{k}(A,\tau)}{k!}e^{-m(A,\tau)}$$ where $m(\cdot,\tau)$ is finitely additive positive mesure on  $Q$ for fixed $\tau$.

This task was offered in \[1\] as the mathematical model of practice problem. The authores in article \[2\] investigated case when initial number and positions of diffusion processes are defined by determinate limited measure  $N(B,\tau)$. Where the  $N(B,\tau)$  is equal to number of points  $x_{k}$  in a set  $B$  and  $N=N(Q,\tau)<\infty$ for fixed $\tau>0$.

We consider the following case

$a(t,x)=a=(\underbrace{0,\dots,0}_{d}),\quad b_{i}(t,x)=b_{i}=
(b_{i1},\dots,b_{id}),~~1\leq i\leq d;$

We define matrix  $\sigma=B^{T}B,\quad B=(b_{ij}),~1\leq i,j\leq d$

$\sigma=(\sigma_{ij}),1\leq i,j\leq d$ and differential operator $A:\sum\limits_{1\leq i,j\leq d}\sigma_{ij}\frac{\partial^{2}}
{\partial x_{i}\partial x_{j}}.$

Let $\sigma$ be a matrix with the following property

$$\sum\limits_{1\leq i,j\leq d}\sigma_{ij}z_{i}z_{j}
\geq \mu |\vec z|^{2}.$$

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

This operator acts in the following space

$$H_{A}=\{u: u\in L_{2}(Q)\cap Au\in L_{2}(Q)\cap u(\partial Q)=0\}$$

with inner product $(u,v)_{A}=(Au,v)$.Here $(,)$ is inner product in $L_{2}(Q)$. The operator  $A$  is positive operator.

It is known \[3\] that the following eigenvalues problem

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

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

$$0<\lambda_{1}<\lambda_{2}<\cdots<\lambda_{s}<\cdots.$$

The corresponding eigenfunctions

$$f_{11},\dots,f_{1n_{1}},\cdots,f_{s1},\dots,f_{sn_{s}},\cdots$$

form 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 $\lambda_{k}$.

We denote by  $\eta(\tau)$  the number of remaining processes in the region  $D$  at time instant  $\tau$.

We also assume that $\sigma$-additive measure $\nu$ is given on the $\Sigma_{\nu}$- algebra sets of

$Q,\quad \nu(Q)<\infty.$ All eigenfunctions $f_{ij}:Q\to R^{1}$ and all measures $m(\cdot,\tau)$  are $(\Sigma_{\nu},\Sigma_{Y})$ measurable. Here $\Sigma_{Y}$ is system of Borel sets of $R^{1}$. Let $\Rightarrow$ denotes the weak convergence of random values.

Put

$$g(\tau)=\exp\left(-\frac{\tau}{2}\lambda_{1}\right).$$

We consider the following initial-boundary problem

$$\frac{\partial u}{\partial t}=\frac{1}{2}\sum\limits_{1\leq i,j\leq d}
\sigma_{ij}\frac{\partial^{2}u}{\partial x_{i}\partial x_{j}}\quad x\in Q;$$

$$u(0,x)= 1\quad\hbox{if}\quad x\in Q;$$

$$u(t,x)=0 \quad \hbox{if}\quad x\in\partial Q,~~~t\geq 0\eqno(2)$$

It is known \[4\], that   $u(\tau, x)$  is equal to probability of remaining in the region  $Q$  at time instant  $\tau$  of a diffusion process from (1) which occurs at the point $(0,x)$ at the initial moment ( $\xi(0)= x,~~  x\in Q$). We designate through $\gamma_{k}=(x^{k}_{1}, \cdots ,x_{d}^{k})$ the initial position of $k$-th process. We define the value of $u(\tau,\gamma_{k})$.

We define a particular solution of (2) in form

$$u(t, x)= u_{1}(t)u_{2}(x).$$

The ordinary argumentaion leads to definition of joined constant $\lambda$:

$$2\frac{1}{u_{1}}\frac{\partial u_{1}}{\partial t}
=\frac{A u_{2}}{u_{2}}=- \lambda.$$

We obtain the following system of tasks due the latter one

$$A u_{2}= -\lambda u_{2};\quad u_{2}(\partial Q)=0.\eqno(3)$$

$$\frac{\partial u_{1}}{\partial t}=
-\frac{\lambda}{2} u_{1};
\quad
u_{1}(0)= 1\eqno(4),$$

It is clear that $u_{1}(t,\lambda)=
\exp(-\frac{t}{2}\lambda )$ is solution of (5) . The soluton of (3) was described above. We assume that system of functions $\{f_{ij}( x), i\geq 1, 1\leq j\leq n_{i}\}$ is orthonormalized with respect to space $L_{2}^{0}(Q)$.

The general solution of problem (2) has the following form

$$u(t,x)=\sum\limits_{j=1}^{\infty}
\exp(-\frac{t}{2}\lambda_{j} )
\sum\limits_{m=1}^{n_{j}}c_{jm}f_{jm}( x)$$

where coefficients $c_{jm}$ are equal to coefficients of decomposition of initial value (unit) by system of functions $f_{jm}$: $c_{jm}=\int\limits_{Q}f_{jm}( x)d x$. The Parseval - Steklov equality is true for these coefficients:

$$\sum\limits_{j=1}^{\infty}\sum\limits_{m=1}^{n_{j}}c_{jm}^{2}=|Q|.\eqno(5)$$

Put $F( x)=\sum\limits_{m=1}^{n_{1}}c_{1i}f_{1m}( x)$. The function  $F(x)$  is continuous and bounded function on the $\bar Q$. Since $u(t,x)$ is probability,it is not diffucult to show that $F( x)\geq 0$ for all $x\in Q$. Let $M=\sup\limits_{ x\in Q}F(x)$. We introduce the following sets

$$B_{k,n}=\{  x\in Q:  \frac{Mk}{n}< F( x)\leq\frac{M(k+1)}{n} \}$$

Here  $0\leq k\leq n-1$  and $n>1$.

Let us denote by $\zeta_{k,n}(\tau), ~1\leq~k\leq n$  the number of unabsorbed processes at time instant  $\tau$ which occur in the region  $B_{k,n}$ at the initial time. These values are independent in totality by assumption. The distribution function of $\zeta_{k,n}(\tau)$  is defined by the following formula

$$P(\zeta_{k,n}(\tau)=l)
=\sum\limits_{d=l}^{\infty}P(\mu(B_{k,n},\tau)=d)\times$$ $$\times\sum\limits_{1\leq i_{1},\cdots,i_{l}\leq d,i_{m}\ne i_{j},m\ne j}
\prod\limits_{k=1}^{l}u(\tau,\gamma_{i_{k}})
\prod\limits^{d}_{s=l+1, i_{s}\notin (i_{1},\cdots,i_{l}),i_{m}\ne i_{j}}
(1- u(\tau,\gamma_{i_{s}})),\quad l=0,1,\dots.$$

Here  $x_{i_{j}}\in B_{k,n}$.

We set

$$a_{k,n}(\tau)=\min\limits_{ x\in B_{k,n}}u(\tau, x),\quad
\bar a_{k,n}(\tau)=1- a_{k,n}(\tau);$$ $$b_{k,n}(\tau)=\max\limits_{ x\in B_{k,n}}u(\tau, x),\quad
\bar b_{k,n}(\tau)=1- b_{k,n}(\tau).$$

Now

$$J_{k,n}(l,\tau):=
\sum\limits_{d=l}^{\infty}\frac{m^{d}(B_{k,n},\tau)}{d!}
\exp(- m(B_{k,n},\tau))
 C_{d}^{l}a^{l}_{k,n}(\tau)\bar b^{d-l}_{k,n}(\tau)\leq$$ $$\leq P(\zeta_{k,n}(\tau)=l)\leq$$ $$\sum\limits_{d=l}^{\infty}
\frac{m^{d}(B_{k,n},\tau)}{d!}\exp(- m(B_{k,n},\tau))
 C_{d}^{l}b^{l}_{k,n}(\tau)\bar a^{d-l}_{k,n}(\tau)=:
I_{k,n}(l,\tau).\eqno(6)$$

Further

$$J_{k,n}(l,\tau)=
\frac{\left(m(B_{k,n},\tau)a_{k,n}(\tau)\right)^{l}}{l!}
\exp(- m(B_{k,n},\tau))
\sum\limits_{d=l}^{\infty}
\frac{\left(\bar b_{k,n}(\tau)m(B_{k,n},\tau)\right)^{d-l}}{(d-l)!}
=$$ $$=\frac{\left(m(B_{k,n},\tau)a_{k,n}(\tau)\right)^{l}}{l!}
\exp(- b_{k,n}(\tau)m(B_{k,n},\tau));$$

By analogy:

$$I_{k,n}(l,\tau)=
\frac{\left(m(B_{k,n},\tau)b_{k,n}(\tau)\right)^{l}}{l!}
\exp(- a_{k,n}(\tau)m(B_{k,n},\tau)).\eqno(7)$$

We introduce the following generating functions

$$\varphi(\tau,s)=\sum\limits_{l\geq 0}s^{l}P(\eta(\tau)=l).$$ $$\varphi_{k,n}(\tau,s)=\sum\limits_{l\geq 0}s^{l}P(\zeta_{k,n}(\tau)=l),
\quad k=\overline{0,n-1},\quad 0\leq s\leq 1.$$

By the construction, $\eta(\tau)$  can be represented in the form  $\eta(\tau)=\zeta_{1,n}+\cdots+\zeta_{n-1,n}(\tau)$.

Thus

$$\varphi(\tau,s)=\prod_{k=0}^{n-1}\varphi_{k,n}(\tau,s).\eqno(8)$$

Combining (6)-(8), we conclude that

$$\exp\{(sa_{k,n}(\tau)-b_{k,n}(\tau))m(B_{k,n},\tau)\}\leq
\varphi_{k.n}(\tau,s)\leq$$ $$\leq \exp\{(sb_{k,n}(\tau)-a_{k,n}(\tau))m(B_{k,n},\tau)\}$$

and

$$\exp\left\{\sum\limits_{k=0}^{n-1}(sa_{k,n}(\tau)-b_{k,n}(\tau))
m(B_{k,n},\tau)\right\}\leq
\varphi(\tau,s)\leq$$ $$\leq
\exp\{\sum\limits_{k=0}^{n-1}(sb_{k,n}(\tau)-a_{k,n}(\tau))
m(B_{k,n},\tau)\}.\eqno(9)$$

Since function  $u(\tau,x)$  is continuous function in $ x\in Q$, there exit a points $x_{*},~~x^{*}\in \bar B_{k,n}$ such that the following equalities have place

$$a_{k,n}(\tau)=
u_{1}(\tau,\lambda_{1})F(x_{*})+
\sum\limits_{k\geq 2}u_{1}(\tau,\lambda_{k})\sum\limits_{m=1}^{n_{k}}c_{km}
f_{km}(x_{*}),$$ $$b_{k,n}(\tau)=
u_{1}(\tau,\lambda_{1})F( x^{*})+
\sum\limits_{k\geq 2}u_{1}(\tau,\lambda_{k})\sum\limits_{m=1}^{n_{k}}c_{km}
f_{km}(x^{*}),$$

here  $ x_{*}:=x_{*}(k,n,\tau),~~x^{*}:=x^{*}(k,n,\tau)$.

Now, we can rewrite the sums in exponentes from (9) in the following forms

$$\sum\limits_{k=0}^{n-1}
\left(sF( x_{*})- F(x^{*})\right)
\exp(-\frac{\tau}{2}\lambda_{1} )
m(B_{k,n},\tau)+$$ $$+
\sum\limits_{k=0}^{n-1}
\exp(-\frac{\tau}{2}\lambda_{1} )
m(B_{k,n},\tau)
\sum\limits_{j\geq 2}
\exp\left(-\frac{\tau}{2}(\lambda_{j}-\lambda_{1})\right)
\sum\limits_{m=1}^{n_{j}}c_{jm}
(sf_{jm}( x_{*})-f_{jm}( x^{*})) , \eqno(10)$$

$$\sum\limits_{k=0}^{n-1}(sF( x^{*})- F( x_{*}))
\exp(-\frac{\tau}{2}\lambda_{1} )
m(B_{k,n},\tau)+$$ $$+\sum\limits_{k=0}^{n-1}
\exp(-\frac{\tau}{2}\lambda_{1} )
m(B_{k,n},\tau)
\sum\limits_{j\geq 2}
\exp\left(-\frac{\tau}{2}(\lambda_{j}-\lambda_{1})\right)
\sum\limits_{m=1}^{n_{j}}c_{jm}
(sf_{jm}( x^{*})- f_{jm}( x_{*})), \eqno(11)$$

We calculate limit of (10) if  $\tau\to\infty$. The first sum of (10) convergers to the following limit under the condition of theorem

$$\sum\limits_{k=0}^{n-1}sF( x_{*})\nu(B_{k.n})
-\sum\limits_{k=0}^{n-1}F( x^{*})\nu(B_{k.n}).$$

This is difference of two integral sums which has the following limit under  $n\to\infty$ (see \[5\]) $$(s-1)\int\limits_{Q}F(x)\nu(d x).$$

Put $$s_{\tau}( x)=
\sum\limits_{j\geq 2}\exp\left(-\frac{\tau}{2}(\lambda_{j}-\lambda_{1})\right)
\sum\limits_{m=1}^{n_{j}}c_{km}
f_{km}(x).$$

We consider sums of eigenfunctions in the form

$$e(x,\lambda)=\sum\limits_{\lambda_{j}\leq \lambda}f^{2}_{jl}(x)$$

The following result is proved in monography \[6,Thm. 17.5.3\] $$\sup\limits_{x\in Q}e(x,\lambda)\leq C \lambda^{\frac{d}{2}}.$$

Asymptotic characteristic of eigenvalues  $\lambda_{j}$  under  $j\to\infty$  is defined by the following inequalities \[3, sec. 18\] $$c_{1}j^\frac{2}{d}\leq \lambda_{j}\leq c_{2} j^\frac{2}{d}, \quad
\hbox{where}\quad c_{1},~c_{2}=const.$$

The latter one, (5) and Caushy-Bunyakovskii inequality lead to the following convergence under $\tau\to\infty$

$$|s_{\tau}( x)|\leq
\sum\limits_{j\geq 2}\exp\left(-\frac{\tau}{2}(\lambda_{j}-\lambda_{1})\right)
\sqrt{\sum\limits_{m=1}^{n_{j}}
c^{2}_{jm}}\sqrt{\sum\limits_{m=1}^{n_{j}}f_{jm}^{2}(x)}\leq$$ $$\leq \sqrt{C}\sum\limits_{j\geq 2}\lambda_{j}^{\frac{d}{2}}\exp(-\frac{\tau}{2}
(\lambda_{j}-\lambda))\sqrt{\sum\limits_{m=1}^{n_{j}}c_{jm}^{2}}\leq$$ $$\leq \sqrt{C} \sqrt{\sum\limits_{j\geq 2} \lambda_{j}^{d}
\exp(-\tau (\lambda_{j}-\lambda_{1}))}
\sqrt{\sum\limits_{j\geq 2}\sum\limits_{m=1}^{n_{j}}c^{2}_{jm}}\to 0.$$

Thus the second sum from(10) convergences to zero.

The similar considerations apply to (11). Proof is complete.

[***Example.***]{} Now we apply the general approach to the particular case.

We consider the case if $Q$ is circle   $Q=\{(x,y): x^{2}+y^{2}\leq r_{0}^{2}\}$. We assume that the diffusion processes occurs at the point $(x_{k},y_{k})\in Q$ at the initial time.

The processes are described in  $Q$  by the following stochastic differential equations

$$d\xi(t)=
\sum\limits_{i=1}^{2}b_{i}dw_{i}(t)\eqno(12)$$ $$\xi(0)=\xi_{0}=(x_{k},y_{k}).$$

where $ b_{1}=(\sigma,0),b_{2}=(0,\sigma) $ and

$W(t)=(w_{i}(t),i=1,2)$ is 2-dimensional Wiener process.

We assume that the equation (12) defines a diffusion process with absorption on the boundary $\partial Q=\{(x,y,z): x^{2}+y^{2}=r_{0}^{2}\}$.

In follows that the $J_{0}(x), J_{1}(x)$ are Bessel functions zero and first order. They are defined as the solutions of the following equations

$$\frac{d^{2} y}{d x^{2}} + \frac{1}{x}\frac{d y}{d x} + (1-\frac{n^{2}}{x^{2}})
=0,$$ $$y(x_{0})=0,~~~  (x_{0}=\sqrt{\lambda}r);\quad  |y(0)|<\infty;$$

under $n=0$ and $n=1$.

The value of $\mu_{m}^{(0)}$ is equal to $m$- th root of the equation $J_{0}(\mu)=0$ \[7,8\].

Let $mes(\cdot)$ denotes the Lebesgue measure.

We set

$$f(\tau):=\exp\left(-\frac{\tau}{2}
\Bigl(\frac{\sigma \mu_{1}^{(0)}}{r}\Bigr)^{2}
\right).$$

We suppose that $m(\cdot,\tau)$ holds the condition $$m(\cdot,\tau)f(\tau) \Rightarrow mes(\cdot) \quad \hbox{if}\quad \tau\to\infty.$$

In this case the system of tasks (3),(4) has the following form

$$\triangle u_{2}= -\mu u_{2}, ~~(x,y)\in C ;\quad
u_{2}(x,y)=0 \quad \hbox{if} \quad x^{2}+y^{2}=r_{0}^{2}\eqno(13),$$ $$\frac{\partial u_{1}}{\partial t}=-\frac{\sigma^{2}}{2} \mu u_{1}, \quad
u_{1}(0)=1.\eqno(14)$$

According to general approach for construction of solution  $u(t,x,y)$  (see,for example, \[7, sec.1V \] we rewrite the task of (13) in polar coordinates: $u_{3}(r,\varphi):=
u_{2}(r\cos\varphi,r\sin\varphi)$. The $u_{3}$ is solution the following problem

$$\frac{\partial^{2} u_{3}}{\partial r^{2}}+\frac{1}{r}\frac{\partial u_{3}}
{\partial r}+ \frac{1}{r^{2}}\frac{\partial^{2}u_{3}}{\partial \varphi^{2}}
+\mu u_{3}=0,$$

$$u_{3}(r_{0},\varphi)=0.$$

We obtain

$$u(t,x,y)=u(t,r)=\sum\limits_{m=1}^{\infty}c_{m}
J_{0}\left(\frac{\mu_{m}^{(0)}}{r_{0}}r\right)
\exp\left(-\frac{t}{2}
\left(\frac{\sigma\mu^{0}_{m}}{r_{0}}\right)^{2}\right),$$

where $c_{m}=2\left(m_{m}^{(0)}J_{1}(\mu_{m}^{(0)})\right)^{-1}$.

The function  $J_{0}\left(\frac{\mu_{1}^{(0)}}{r_{0}}r\right)$  is strictly decreasing function if  $0\leq r\leq r_{0}$ . Thus we can construct the partitions  $B_{k,n}$  by the following partitions $$\tilde B_{k,n}=\left\{ (x,y)\in C: \frac{r_{0}k}{n}< \sqrt{x^{2}+y^{2}}
\leq \frac{r_{0}(k+1)}{n}\right\},\quad 0\leq i\leq n-1.$$

Now  $mes(\tilde B_{k,n})=g(\frac{k+1}{n})-g(\frac{k}{n})$  where    $g(x)=\pi r_{0}^{2}x^{2}, ~~~0\leq x\leq 1$.

Finally, the parameter of Poisson distribution is equal to

$$a=2\left(m_{1}^{(0)}J_{1}(\mu_{1}^{(0)})\right)^{-1}2\pi r_{0}^{2}
\int\limits_{0}^{1}J_{0}(\mu_{1}^{(0)}x)xdx=\pi
\left(\frac{2 r}{\mu_{1}^{(0)}}\right)^{2}.$$

We used the following known relation   $\alpha J_{0}(\alpha)=\left[\alpha J_{1}(\alpha)\right]'$ \[7,p.466\] for calculation of the latter integral.

\[\] V.A.Gasanenko and A.B.Roitman Rarefaction of moving diffusion particles The Ukrainian Mathematical Journal 56 2004 691-694 \[\] A.Fedullo, V.A. Gasanenko Limit theorems for rarefaction of set of diffusion processes by boundaries Theory of Stochastic Processes 2005 (to appear) \[\] S.G. Mihlin Partial differential linear equations Vyshaij shkola Moscow 1977 431 p \[\] I.I.Gikhman, A.V. Skorokhod Introduction to the theory of random processes Nauka Moscow 1977 568 p \[\] A.N. Kolmogorov, S.V. Fomin Elements of theory of functions and functional analysis Nauka Moscow 1972 496 p \[\] L. H$\ddot o$rmander The analysis of Linear Partial Differential Operators III Spinger-Verlag 1985 \[\] A.N.Tikhonov, A.A.Samarsky The equations of mathematical physics Nauka Moskow 1977 736 \[\] E.Janke, F.Emde and F.Losch Special functions Nauka Moskow 1968 344
