Robustness of exponential stability of a class of stochastic functional differential equations with infinite delay

We regard the stochastic functional differential equation with infinite delay d x ( t ) = f ( x t ) d t + g ( x t ) d w ( t ) as the result of the effects of stochastic perturbation to the deterministic functional differential equation x ̇ ( t ) = f ( x t ) , where x t = x t ( θ ) ∈ C ( ( − ∞ , 0 ] ; R n ) is defined by x t ( θ ) = x ( t + θ ) , θ ∈ ( − ∞ , 0 ] . We assume that the deterministic system with infinite delay is exponentially stable. In this paper, we shall characterize how much the stochastic perturbation can bear such that the corresponding stochastic functional differential system still remains exponentially stable.