Computable Convergence Rates for Subgeometrically Ergodic Markov Chains

Computable bounds of ${\ell}^2$-spectral gap for discrete Markov chains with band transition matrices

We analyse the $\ell^2(π)$-convergence rate of irreducible and aperiodic Markov chains with $N$-band transition probability matrix $P$ and with invariant distribution $π$. This analysis is heavily based on: first the study of the essential spectral radius $r\_{ess}(P\_{|\ell^2(π)})$ of $P\_{|\ell^2(π)}$ derived from Hennion's quasi-compactness criteria; second the connection between the Spectral Gap property (SG$\_2$) of $P$ on $\ell^2(π)$ and the $V$-geometric ergodicity of $P$. Specifically, (SG$\_2$) is shown to hold under the condition $ α\_0 := \sum\_{{m}=-N}^N \limsup\_{i\rightarrow +\in…

Approximating Markov chains and V-geometric ergodicity via weak perturbation theory

Let $P$ be a Markov kernel on a measurable space $\X$ and let $V:\X\r[1,+\infty)$. This paper provides explicit connections between the $V$-geometric ergodicity of $P$ and that of finite-rank nonnegative sub-Markov kernels $\Pc_k$ approximating $P$. A special attention is paid to obtain an efficient way to specify the convergence rate for $P$ from that of $\Pc_k$ and conversely. Furthermore, explicit bounds are obtained for the total variation distance between the $P$-invariant probability measure and the $\Pc_k$-invariant positive measure. The proofs are based on the Keller-Liverani perturbat…

Lower estimates of transition densities and bounds on exponential ergodicity for stochastic PDE's

A formula for the transition density of a Markov process defined by an infinite-dimensional stochastic equation is given in terms of the Ornstein--Uhlenbeck bridge and a useful lower estimate on the density is provided. As a consequence, uniform exponential ergodicity and $V$-ergodicity are proved for a large class of equations. We also provide computable bounds on the convergence rates and the spectral gap for the Markov semigroups defined by the equations. The bounds turn out to be uniform with respect to a large family of nonlinear drift coefficients. Examples of finite-dimensional stochast…

Continuous-time block-monotone Markov chains and their block-augmented truncations

This paper considers continuous-time block-monotone Markov chains (BMMCs) and their block-augmented truncations. We first introduce the block monotonicity and block-wise dominance relation for continuous-time Markov chains, and then provide some fundamental results on the two notions. Using these results, we show that the stationary distribution vectors obtained by the block-augmented truncation converge to the stationary distribution vector of the original BMMC. We also show that the last-column-block-augmented truncation (LC-block-augmented truncation) provides the best (in a certain sense) …

A hypocoercivity related ergodicity method with rate of convergence for singularly distorted degenerate Kolmogorov equations and applications

In this article we develop a new abstract strategy for proving ergodicity with explicit computable rate of convergence for diffusions associated with a degenerate Kolmogorov operator L. A crucial point is that the evolution operator L may have singular and nonsmooth coefficients. This allows the application of the method e.g. to degenerate and singular particle systems arising in Mathematical Physics. As far as we know in such singular cases the relaxation to equilibrium can't be discussed with the help of existing approaches using hypoellipticity, hypocoercivity or stochastic Lyapunov type te…

Dynamics of continuous time Markov chains with applications

This paper contributes an in-depth study of properties of continuous time Markov chains (CTMCs) on non-negative integer lattices $\N_0^d$, with particular interest in one-dimensional CTMCs with polynomial transitions rates. Such stochastic processes are abundant in applications, in particular in biology. We characterize the structure of the state space of general CTMCs on $\N_0^d$ in terms of the set of jump vectors and their corresponding transition rate functions. For CTMCs on $\N_0$ with polynomial transition rate functions, we provide threshold criteria in terms of easily computable parame…

连续时间马尔可夫链极限分布的渐近尾部

The asymptotic tails of limit distributions of continuous time Markov chains

本文基于通量平衡方程建立平稳测度与QSD恒等式，用三个参数分类幂律转移速率连续时间马尔可夫链的三类尾部渐近，并推广至多种模型及离散时间情形。

非负整数上具有任意转移速率的马尔可夫链动力学分类及其应用

Classification for dynamics of Markov chains on non-negative integers with arbitrary transition rates and its application

本文对非负整数上具有任意转移速率的马尔可夫链给出基于渐近行为的完整动力学分类，涵盖爆炸性、常返性与指数遍历性，并应用于有理速率（含质量作用、米氏及Haldane动力学）的充要条件刻画。

非可逆马尔可夫链的非渐近混合时间分析

Non Asymptotic Mixing Time Analysis of Non-Reversible Markov Chains

本文提出了一个统一的算子理论框架，通过投影转移算子分析非可逆马尔可夫链的非渐近混合时间，并基于其谱特征建立了卡方散度界限，为动量采样器等非可逆链提供了精确的收敛分析。

BAR-SOT方法：长期平均成本控制作为随机最优自输运

The BAR-SOT Method: Long-term Average Cost Control as Stochastic Optimal Self-Transport

提出BAR-SOT方法，将长期平均成本控制重构为有限时域随机最优自输运问题，通过三种等价表述及神经对偶参数化，在多个基准上超越强化学习基线和优先级启发式。