2025/04/24 by Wojciech Cygan, Cygan, Wojciech, Kamil Kaleta +5
Business, Management and Accounting · Mathematics · #05C81 #35K08 #47D08 #Advanced Queuing Theory Analysis #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Primary: 60J10 #Probability (math.PR) #Secondary: 60J45 #Spectral Theory (math.SP) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2504.17879
openalex publication_date 2025/04/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 un(x,y) of the perturbed chain. These estimates are of the form un(x,y)\asymp λ0nϕ0(x)\widehatϕ0(y)+Fn(x,y), where ϕ0 (and \widehatϕ0) are the (dual) eigenfunctions for the lowest eigenvalue λ0; the perturbation Fn(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 \enquotedirect 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.