Averaging in hyperbolic systems subjected to weakly dependent random disturbances

Abstract

The first initial boundary value problem is considered for a hyperbolic equation with a small parameter for an external action described by some stochastic process satisfying some of the conditions of weak dependence. Averaging of the coefficients over the temporal variable is conducted. The existence is assumed of a unique generalized solution both for the initial stochastic problem and for the problem with an “averaged” equation, which turns out to be deterministic. For the probability of deviation of a solution of the initial equation from the solution of the “averaged” problem, exponential bounds are established of the type of S. N. Bernshtein inequalities for the sums of independent random variables.