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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. 

This paper focuses on two-time-scale coupled stochastic functional differential equations (SFDEs). The system under consideration has a slow component and a fast component. Both components depend on the segment process (an infinite dimension process) of the slow component. To overcome the difficulty due to the past dependence and the coupling of the segment process, such properties as the H\"{o}lder continuity and tightness on a space of continuous functions are investigated first for the segment process. In addition, it is also shown that the solution of a fixed-x equation depends continuously on the parameters. Then using the martingale problem formulation, an average principle is established by a direct averaging. 

This paper aims to study stability in distribution of Markovian switching jump diffusions. The main motivation stems from stability and stabilizing hybrid systems in which there is no trivial solution. An explicit criterion for stability in distribution is derived. The stabilizing effects of Markov chains, Brownian motions, and Poisson jumps are revealed. Based on these criteria, stabilization problems of stochastic differential equations with Markovian switching and Poisson jumps are developed.