Asymptotic Bismut formulae for stochastic functional differential equations with infinite delay

Abstract

Using Malliavin calculus, this paper establishes asymptotic Bismut formulae for stochastic functional differential equations with infinite delay. Both nondegenerate and degenerate diffusion coefficients are treated. In addition, combined with the corresponding exponential ergodicity, stabilization bounds for ∇ P t f \nabla P_{t}f as t → ∞ t\rightarrow \infty are derived.