On large deviations of solutions of nonlinear stochastic equations

Abstract

This paper proves the existence and uniqueness of a strong solution of quasilinear parabolic partial differential equations with white noise. It is proved that the solutions continuously depend on the trajectories of the Wiener process. The main result is exponential estimates for the probabilities of large deviations of the solutions of quasilinear parabolic equations with white noise. These probabilities are estimated via the action functional. Estimates of two types are established, a lower bound on the probability that the solution is in a neighborhood of a fixed trajectory and an upper bound on the probability of large deviation of the solution from the set of trajectories with bounded action. These results generalize the estimates established by Venttsel' and Freidlin for ordinary differential equations to the case of parabolic partial differential equations.