Averaging in parabolic systems subject to weakly dependent random actions. The L2-approach

The first initial-boundary problem for a parabolic equation with a small parameter under external action described by some random process satisfying an arbitrary condition of weak dependence is considered. Averaging of the coefficients over a time variable is carried out. The existence of a generalized solution for the initial stochastic problem as well as for the problem with an averaged equation which turns out to be deterministic is assumed. Exponential bounds of the type of the well-known Bernstein inequalities for a sum of independent random variables are established for the probability of the deviation of the solution of the initial equation from the solution of the averaged problem.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 3, pp. 315–322, March, 1991.