Averaging in stochastic quasi-linear parabolic equations. (Russian) Zbl 0705.60049

Bondarev, B. V.

Teor. Veroyatn. Mat. Stat., Kiev 41, 16-23 (1989).

The following stochastic partial differential equation is studied: \[ (1)\quad d_ t\xi_{\epsilon}(t,x)= \epsilon[L\xi_{\epsilon}(t,x)+ A(t,x,\xi_{\epsilon}(t,x))dt+ \sum^{n}_{i=1}\sigma_ i(t,x,\xi_{\epsilon}(t,x))dW_ i(t)], \] \(\xi_{\epsilon}(t,x)|_{t=0}= \phi(x)\), where \[ L U(t,x)=\sum^{n}_{i=1} a_{ij}(t,x)\partial^ 2u/\partial x_ i\partial x_ j+ \sum^{n}_{i=1}b_ i(t,x)\partial u/\partial x_ i+ C(t,x)U(t,x). \] The averaging of the coefficients \(a_{ij}(t,x)\), \(b_ i(t,x)\), \(C(t,x)\) and \(A(t,x,z)\) yields the corresponding Cauchy equation. The inequality for the probability of the fluctuation of the solution of (1) from the solution of the averaged problem is obtained.

Reviewer: D.Jaruskova

MSC:

60H15

Stochastic partial differential equations (aspects of stochastic analysis)

Keywords:

parabolic operator; averaging principle; Cauchy equation