A Markov jump process approximation of the stochastic Burgers equation

The Augmented Jump Chain -- a sparse representation of time-dependent Markov jump processes

Modern methods of simulating molecular systems are based on the mathematical theory of Markov operators with a focus on autonomous equilibrated systems. However, non-autonomous physical systems or non-autonomous simulation processes are becoming more and more important. We present a representation of non-autonomous Markov jump processes as autonomous Markov chains on space-time. Augmenting the spatial information of the embedded Markov chain by the temporal information of the associated jump times, we derive the so-called augmented jump chain. The augmented jump chain inherits the sparseness o…

关于具有流之间随机切换的分段确定性 Markov 过程相关不变测度的绝对连续性

On absolute continuity of invariant measures associated with a piecewise-deterministic Markov processes with random switching between flows

本文研究了生物模型中分段确定性 Markov 过程平稳分布的绝对连续性，该过程由 Poisson 过程跳跃和连续半流切换组成，提出了一组可验证条件，证明其不变分布具有关于 Lebesgue 测度的密度。

Space-time stationary solutions for the Burgers equation

We construct space-time stationary solutions of the 1D Burgers equation with random forcing in the absence of periodicity or any other compactness assumptions. More precisely, for the forcing given by a homogeneous Poissonian point field in space-time we prove that there is a unique global solution with any prescribed average velocity. These global solutions serve as one-point random attractors for the infinite-dimensional dynamical system associated to solutions to the Cauchy problem. The probability distribution of the global solutions defines a stationary distribution for the corresponding …

Large deviations theory for Markov jump models of chemical reaction networks

We prove a sample path Large Deviation Principle (LDP) for a class of jump processes whose rates are not uniformly Lipschitz continuous in phase space. Building on it we further establish the corresponding Wentzell-Freidlin (W-F) (infinite time horizon) asymptotic theory. These results apply to jump Markov processes that model the dynamics of chemical reaction networks under mass action kinetics, on a microscopic scale. We provide natural sufficient topological conditions for the applicability of our LDP and W-F results. This then justifies the computation of non-equilibrium potential and expo…

Stabilization of continuous-time Markov/semi-Markov jump linear systems via finite data-rate feedback

This paper investigates almost sure exponential stabilization of continuous-time Markov jump linear systems (MJLSs) under communication data-rate constraints by introducing sampling and quantization into the feedback control. Different from previous works, the sampling times and the jump times are independent of each other in this paper. The quantization is recursively adjusted on the sampling time, and its updating strategy does not depend on the switching in a sampling interval. In other words, the explicit value of the switching signal in a sampling interval is not necessary. The numericall…

鲁棒周期稳定性蕴含马尔可夫跳跃线性系统的均匀指数稳定性及随机线性常微分方程

Robust periodic stability implies uniform exponential stability of Markovian jump linear systems and random linear ordinary differential equations

本文证明线性cocycle鲁棒周期稳定性蕴含均匀稳定性，涉及马尔可夫跳跃线性系统和随机线性常微分方程的稳定性分析。