A note on almost sure exponential stability for stochastic partial functional differential equations

In a recent paper, Taniguchi (Stochastic Anal. Appl. 16 (5) (1998) 965-975) investigated the almost sure exponential stability of the mild solutions of a class of stochastic partial functional differential equations. Precisely, as small delay interval assumption is imposed, sufficient conditions are obtained there to ensure the almost sure exponential stability of the mild solutions of the given stochastic systems. Unfortunately, the main results derived by him are somewhat restrictive to be applied for practical purposes. In the note we shall prove that for a class of stochastic functional differential equations the small delay interval assumption imposed there is actually unnecessary and can be removed.