Feller property and exponential ergodicity of diffusion processes with state-dependent switching

In this paper we consider the Feller property and the exponential ergodicity for general diffusion processes with state-dependent switching. We prove their Feller continuity by means of intro- ducing some auxiliary processes and by making use of the Radon-Nikodym derivatives. Furthermore, we also prove their strong Feller continuity and their exponential ergodicity under some reasonable conditions.

Exponential and Strong Ergodicity for Markov Processes with an Application to Queues

For an ergodic continuous-time Markov process with a particular state in its space,the authors provide the necessary and sufficient conditions for exponential and strong ergodicity in terms of the moments of the first hitting time on the state.An application to the queue length process of M/G/1 queue with multiple vacations is given.

Random Walk Approximation for Irreversible Drift-Diffusion Process on Manifold:Ergodicity,Unconditional Stability and Convergence

Irreversible drift-diffusion processes are very common in biochemical reactions.They have a non-equilibrium stationary state(invariant measure)which does not satisfy detailed balance.For the corresponding Fokker-Planck equation on a closed manifold,using Voronoi tessellation,we propose two upwind finite volume schemes with or without the information of the invariant measure.Both schemes possess stochastic Q-matrix structures and can be decomposed as a gradient flow part and a Hamiltonian flow part,enabling us to prove unconditional stability,ergodicity and error estimates.Based on the two upwind schemes,several numerical examples–including sampling accelerated by a mixture flow,image transformations and simulations for stochastic model of chaotic system–are conducted.These two structurepreserving schemes also give a natural random walk approximation for a generic irreversible drift-diffusion process on a manifold.This makes them suitable for adapting to manifold-related computations that arise from high-dimensional molecular dynamics simulations.

Perturbation analysis for continuous-time Markov chains

We investigate perturbation for continuous-time Markov chains(CTMCs) on a countable state space. Explicit bounds on ?D and D are derived in terms of a drift condition, where ? and D represent the perturbation of the intensity matrices and the deviation matrix, respectively. Moreover, we obtain perturbation bounds on the stationary distributions, which extends the results by Liu(2012) for uniformly bounded CTMCs to general(possibly unbounded) CTMCs. Our arguments are mainly based on the technique of augmented truncations.

Stochastic Liénard Equations with State-Dependent Switching

This work focuses on stochastic Lienard equations with state-dependent switching. First, the existence and uniqueness of a strong solution are obtained by successive construction method. Next, strong Feller property is proved by introducing certain auxiliary processes and using the Radon-Nikodym derivatives and truncation arguments. Based on these results, positive Harris recurrence and exponential ergodicity are obtained under the Foster-Lyapunov drift conditions. Finally, examples using van der Pol equations are presented for illustrations, and the corresponding Foster-Lyapunov functions for the examples are constructed explicitly.

Markov branching processes with immigration-migration and resurrection

We consider a modified Markov branching process incorporating with both state-independent immigration-migration and resurrection. The effect of state-independent immigration-migration is firstly in- vestigated in detail. The explicit expressions for the extinction probabilities and mean extinction times are presented. The ergodicity and stability properties of the process incorporating with resurrection structure are then investigated. The conditions for recurrence, ergodicity and exponential ergodicity are obtained. An explicit expression for the equilibrium distribution is also presented. As a preparation, the criteria for regularity and uniqueness for such structure are firstly established.