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On a problem with the small parameter for a class of stochastic processes. (Russian) Zbl 0717.60078

Statistics and control of stochastic processes, Collect. Artic., Moskva, 191-195 (1989).
[For the entire collection see Zbl 0688.00011.]
The asymptotic behaviour of the solutions of the system of partial differential equations \[ \partial u^{\epsilon}_ k(t,x)/\partial t=\epsilon^{-1}L_ ku_ k+\sum^{n}_{j=1}c_{kj}(x)u_ j,\quad t>0,\quad x\in D\subset {\mathbb{R}}^ r,\quad k=1,...,n,\quad u_ k(0,x)=g_ k(x) \] and with Neumann boundary conditions when \(\epsilon\) tends to zero is studied. Here \(L_ k\) are elliptic operators. The limit process \(u_ k(t)\) and the deviation to the order o(\(\epsilon\)) of the \(u^{\epsilon}_ k(t,x)\) from \(u_ k(t)\) using the description of \(u^{\epsilon}_ k(t,x)\) as a mean value of some functional of a Markov process are described.
A theorem for the case when the factor \(\epsilon^{-1}\) stands in front of one \(L_ k\) only is also formulated.
Reviewer: P.Holicky

MSC:

60H30 Applications of stochastic analysis (to PDEs, etc.)
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J25 Continuous-time Markov processes on general state spaces

Citations:

Zbl 0688.00011