On the Small-Time Asymptotics of Solutions of Linear and Nonlinear Stochastic Differential Equations

Abstract

This chapter gives a survey of some results of the small-time asymptotics of solutions of stochastic differential equations on Hilbert spaces, which include solutions of some simple stochastic partial differential equations. It discusses the Gaussian cases, and treats more general diffusions than Ornstein-Uhlenbeck processes. The operator function is smooth enough so that the diffusion is a solution of a stochastic differential equation on the Hilbert space. The chapter describes that the heat kernel is the transition density of the Brownian motion. Much work has been done to extend the asymptotics to more general situations where the Laplacian is replaced by general elliptic operators.