Moment and almost sure Lyapunov exponents of mild solutions of stochastic evolution equations with variable delays via approximation approaches

Abstract

Several criteria for the asymptotic exponential stability of a class of Hilbert space-valued, non-autonomous stochastic evolution equations with variable delays are presented. This formulation is particularly suitable for the treatment of mild solutions of general stochastic delay partial differential equations. The principal technique of our investigation is to construct a proper approximating strong solution system and carry out a limiting type of argument to obtain the required exponential stability. As a consequence, stability results from A. Ichikawa [8] are generalized to cover a class of non-autonomous stochastic delay evolution equations. In particular, we improve the recent results of T. Taniguchi [14] to remove the time delay interval restriction there.