This paper is a continuation of the study on the stability speed for Markov processes. It extends the previous study of the ergodic convergence speed to the non-ergodic one, in which the processes are even allowed to be explosive or to have general killings. At the beginning stage, this paper is concentrated on the birth-death processes. According to the classification of the boundaries, there are four cases plus one more having general killings. In each case, some dual variational formulas for the convergence rate are presented, from which, the criterion for the positivity of the rate and an approximating procedure of estimating the rate are deduced. As the first step of the approximation, the ratio of the resulting bounds is usually no more than 2. The criteria as well as basic estimates for more general types of stability are also presented. Even though the paper contributes mainly to the non-ergodic case, there are some improvements in the ergodic one. To illustrate the power of the results, a large number of examples are included.

Speed of stability for birth-death processes

This paper is a continuation of the study on the stability speed for Markov processes. It extends the previous study of the ergodic convergence speed to the non-ergodic one, in which the processes are even allowed to be explosive or having general killings. At the beginning stage, this paper is concentrated on the birth-death processes. According to the classification of the boundaries, there are four cases plus one having general killings. In each case, some dual variational formulas for the convergence rate are presented, from which the criterion for the positivity of the rate and an approximating procedure of estimating the rate are deduced. As the first step of the approximation, the ratio of the resulting bounds is usually no more than 2. The criteria as well as basic estimates for more general types of stability are also presented. Even though the paper contributes mainly to the non-ergodic case, there are some improvements in the ergodic one. To illustrate the power of the results, a large number of examples are included.

Speed of stability for birth--death processes

On shift Harnack inequalities for subordinate semigroups and moment estimates for Lévy processes

Gradient estimates for SDEs driven by multiplicative Lévy noise

Gradient estimates are derived, for the first time, for the semigroup associated to a class of stochastic differential equations driven by multiplicative L\'evy noise. In particular, the estimates are sharp for $\alpha$-stable type noises. To derive these estimates, a new derivative formula of Bismut-Elworthy-Li's type is established for the semigroup by using the Malliavin calculus and a finite-jump approximation argument.

Jian Wang

Papers

Harnack inequalities for stochastic equations driven by Levy noise

Functional inequalities for transient birth–death processes and their applications

Poincaré-type inequalities for singular stable-like Dirichlet forms