标题： 热核、固有收缩性和受势能杀死的离散时间马尔可夫链的遍历性 显示英文标题

标题： Heat kernels, intrinsic contractivity and ergodicity of discrete-time Markov chains killed by potentials

摘要： 我们研究了定义在可数无穷状态空间上的离散时间Markov链，这些链受到相当广义的约束势（即在无穷远处增长的势）的扰动。利用经典Feynman-Kac公式的离散时间类比，我们得到了受扰链的$n$步热核$u_n(x,y)$的双边估计。 这些估计的形式为$u_n(x,y)\asymp \lambda_0^n\phi_0(x)\widehat\phi_0(y)+F_n(x,y)$，其中$\phi_0$（和$\widehat\phi_0$）是对应最小特征值$\lambda_0$的（对偶）特征函数；扰动$F_n(x,y)$明确给出，若$x$或$y$属于有界集，则该扰动消失。 关键假设是链是均匀懒惰的，并且满足\enquote{direct step property}（DSP）。这意味着链从状态$x$转移到状态$y$的可能性在单步内比在两步或多步内更大。 从热核估计的形式出发，我们定义了内在（或基态变换）链，并引入了与时间相关的超压缩性概念——渐近和进展性内在超压缩性——这些可以与限制势的增长行为联系起来；这使得我们可以考虑任意缓慢增长的势。 这些新的超压缩性概念还导致了受扰动和基态变换马尔可夫链一致（拟-）遍历性的特征化。 在本文结尾，我们给出了多个例子，以说明我们的发现如何与现有模型相关，例如无限图上的最近邻随机游走、次序过程或非可逆马尔可夫链。 显示英文摘要

摘要： We study discrete-time Markov chains on countably infinite state spaces, which are perturbed by rather general confining (i.e.\ growing at infinity) potentials. Using a discrete-time analogue of the classical Feynman--Kac formula, we obtain two-sided estimates for the $n$-step heat kernels $u_n(x,y)$ of the perturbed chain. These estimates are of the form $u_n(x,y)\asymp \lambda_0^n\phi_0(x)\widehat\phi_0(y)+F_n(x,y)$, where $\phi_0$ (and $\widehat\phi_0$) are the (dual) eigenfunctions for the lowest eigenvalue $\lambda_0$; the perturbation $F_n(x,y)$ is explicitly given, and it vanishes if either $x$ or $y$ is in a bounded set. The key assumptions are that the chain is uniformly lazy and that the \enquote{direct step property} (DSP) is satisfied. This means that the chain is more likely to move from state $x$ to state $y$ in a single step rather than in two or more steps. Starting from the form of the heat kernel estimate, we define the intrinsic (or ground-state transformed) chains and we introduce time-dependent ultracontractivity notions -- asymptotic and progressive intrinsic ultracontractivity -- which we can link to the growth behaviour of the confining potential; this allows us to consider arbitrarily slow growing potentials. These new notions of ultracontractivity also lead to a characterization of uniform (quasi-)ergodicity of the perturbed and the ground-state transformed Markov chains. At the end of the paper, we give various examples that illustrate how our findings relate to existing models, e.g.\ nearest-neighbour walks on infinite graphs, subordinate processes or non-reversible Markov chains.