On the limit behavior of solutions of systems of stochastic diffusion equations. (English. Russian original) Zbl 0699.60042

Kharkova, M. V.

Theory Probab. Math. Stat. 40, 145-150 (1990); translation from Teor. Veroyatn. Mat. Stat., Kiev 40, 121-126 (1989).

Summary: The system of Itô stochastic diffusion equations \[ (1)\quad d\xi_ n(t)=a_ n(t,\xi_ n(t))dt+\sum^{m}_{j=1}\sigma^ j_ n(t,\xi_ n(t))d\omega^ j_ n(t) \] is considered, where n is a parameter, \(a_ n(t,x)\) and \(\sigma^ j_ n(t,x)\), \(j=1,...,m\), are vector-valued functions, \(x=(x^ i\), \(i=1,...,m)\) is a point in \(R^ m\), \(t\in [0,1]\), and the \(\omega^ j_ n(t)\) are jointly independent one- dimensional Wiener processes.
The limit behavior of \(\xi_ n(t)\) as \(n\to \infty\) is investigated for the class of equations (1) in which the inner product \((x,a_ n(t,x))\leq C\) is uniformly bounded with respect to x and n, and the coefficient \(a_ n(t,x)\) is allowed to be unbounded with respect to n in certain domains.

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60J60	Diffusion processes
60F05	Central limit and other weak theorems

Keywords:

limit behavior of solutions; stochastic diffusion equations; Wiener processes