Ergodicity Of Stochastic Differential Equations

In this paper,we gave an simple proof of that the strong Feller property of general Markov semigroups in the distance space implies the disjointedness of topological sup-ports of different ergodic measures.Thus,the requirements of the irreducability of the Markov process in proving the uniqueness of the invariant measure can be reduced.After that,we studied the ergodicity of stochastic differential equations driven by Levy noise with local Lipschitz coefficient and gave some examples.The result is ap-propriate for applying in the stochastic dynamic systems with polynomial growth coef-ficients.At last,using Malliavain matrix,under invertible and nun-invertiable condition,we obtained the gradient estimation of relevant Markovian semigroup and proved that when Malliavain matrix is invertiable,the relevant Markovian semigroup has strong Feller property.