Exponential Ergodicity Of Non-Lipschitz Stochastic Differential Equations

In this paper we consider the exponential ergodicity for non-Lipschitz stochastic differential equations, and it has been widely studied especially when the coefficient has continuous mode. An important step in handling this kind of problem is to prove the existence and uniqueness of an invariant measure. In recent years, through methods like random homeomorphism flow, large deviations, and Euler approximation approach, scholars has obtained many important results. For example, Zhang [34] in his article used the coupling method to prove that kind of equation with non-Lipschitz coefficients has strong Feller property and owns irreducibility, the key of the proof is to construct the coupling function properly. He also proved the exponential ergodicity property whence satisfing one of the usual Lyapunov conditions on the coefficients.Based on the study of Zhang, we studied the exponential ergodicity property of non-lipschitz stochastic differential equations with jumps(driven by Brown motion and Poisson process)and we listed the sufficient conditions in the proof. Finally, we also considered the exponential ergodicity of time dependent stochastic differential equations without jumps.