Jianhai Bao

Permanence and extinction of regime-switching predator-prey models

In this work we study the permanence and extinction of a regime-switching predator-prey model with the Beddington--DeAngelis functional response. The switching process is used to describe the random changes of corresponding parameters such as birth and death rates of a species in different environments. When a prey will die out in some fixed environments and will not in others, our criteria can justify whether it dies out in a random switching environment. Our criteria are rather sharp, and they cover the known on-off type results on permanence of predator-prey models without switching. Our method relies on the recent study of ergodicity of regime-switching diffusion processes.

Stability in distribution of mild solutions to stochastic partial differential equations

In the present paper, we investigate stochastic partial differential equations. By introducing a suitable metric between the transition probability functions of mild solutions, we derive sufficient conditions for stability in distribution of mild solutions. Consequently, we generalize some existing results to infinite dimensional cases. Finally, one example is constructed to demonstrate the applicability of our theory.

Large deviations for neutral functional SDEs with jumps

In this paper, by the weak convergence method, based on a variational representation for positive functionals of a Poisson random measure and Brownian motion, we establish uniform large deviation principles (LDPs) for a class of neutral stochastic differential equations driven by jump processes. As a byproduct, we also obtain uniform LDPs for neutral stochastic differential delay equations which, in particular, allow the coefficients to be highly nonlinear with respect to the delay argument.

DERIVATIVE FORMULA AND HARNACK INEQUALITY FOR DEGENERATE FUNCTIONAL SDEs

By constructing successful couplings, the derivative formula, gradient estimates and Harnack inequalities are established for the semigroup associated with a class of degenerate functional stochastic differential equations.

Two-time-scale stochastic partial differential equations driven by

This paper focuses on stochastic partial differential equations (SPDEs) under two-time-scale formulation. Distinct from the work in the existing literature, the systems are driven by

\alpha

\alpha

-stable noises: Averaging principles

-stable processes with

Transportation Cost Inequalities for Neutral Functional Stochastic Equations

\alpha \in(1,2)

Stationary distributions for retarded stochastic differential equations without dissipativity

. In addition, the SPDEs are either modulated by a continuous-time Markov chain with a finite state space or have an addition fast jump component. The inclusion of the Markov chain is for the needs of treating random environment, whereas the addition of the fast jump process enables the consideration of discontinuity in the sample paths of the fast processes. Assuming either a fast changing Markov switching or an additional fast-varying jump process, this work aims to obtain the averaging principles for such systems. There are several distinct difficulties. First, the noise is not square integrable. Second, in our setup, for the underlying SPDE, there is only a unique mild solution and as a result, there is only mild It\^{o}s formula that can be used. Moreover, another new aspect is the addition of the fast regime switching and the addition of the fast varying jump processes in the formulation, which enlarges the applicability of the underlying systems. To overcome these difficulties, a semigroup approach is taken. Under suitable conditions, it is proved that the

Exponential ergodicity for retarded stochastic differential equations

p