Stability And Ergodicity Of A Class Of Pure Jump Processes

In this paper,we first study the Maximum principle of pure jump stable processes,and use Dirichlet equations,Lyapunov functions to explore the regularity of solution of corresponding stochastic differential equation.Secondly,referring to the Harnack inequality corresponding to pure jump processes,the conditions for the existence of the Harnack inequality of stochastic differential equation driven by the pure jump stable processes are obtained.Finally,the application of pure jump stable processes in biological population is discussed,and on the basis of the Maximum principle,the sufficient conditions for the existence and unique stationary distribution of the corresponding solution process of the studied stochastic differential equation are proved by using Lyapunov functions and Khasminskii lemma.