Small Time Asymptotics for Stochastic Evolution Equations

Abstract

We obtain a large deviation principle describing the small time asymptotics of the solution of a stochastic evolution equation with multiplicative noise. Our assumptions are a condition on the linear drift operator that is satisfied by generators of analytic semigroups and Lipschitz continuity of the nonlinear coefficient functions. Methods originally used by Peszat (Probab. Theory Relat. Fields 98:113–136, 1994) for the small noise asymptotics problem are adapted to solve the small time asymptotics problem. The results obtained in this way improve on some results of Zhang (Ann. Probab. 28(2):537–557, 2000).