Heat kernels, intrinsic contractivity and ergodicity of discrete-time Markov chains killed by potentials
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| Format: | Preprint |
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2025
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| _version_ | 1866917998695022592 |
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| author | Cygan, Wojciech Kaleta, Kamil Schilling, René L. Śliwiński, Mateusz |
| author_facet | Cygan, Wojciech Kaleta, Kamil Schilling, René L. Śliwiński, Mateusz |
| contents | 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 $u_n(x,y)\asymp λ_0^nϕ_0(x)\widehatϕ_0(y)+F_n(x,y)$, where $ϕ_0$ (and $\widehatϕ_0$) are the (dual) eigenfunctions for the lowest eigenvalue $λ_0$; 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_17879 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Heat kernels, intrinsic contractivity and ergodicity of discrete-time Markov chains killed by potentials Cygan, Wojciech Kaleta, Kamil Schilling, René L. Śliwiński, Mateusz Probability Mathematical Physics Dynamical Systems Functional Analysis Spectral Theory Primary: 60J10, Secondary: 60J45, 05C81, 35K08, 47D08 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 $u_n(x,y)\asymp λ_0^nϕ_0(x)\widehatϕ_0(y)+F_n(x,y)$, where $ϕ_0$ (and $\widehatϕ_0$) are the (dual) eigenfunctions for the lowest eigenvalue $λ_0$; 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. |
| title | Heat kernels, intrinsic contractivity and ergodicity of discrete-time Markov chains killed by potentials |
| topic | Probability Mathematical Physics Dynamical Systems Functional Analysis Spectral Theory Primary: 60J10, Secondary: 60J45, 05C81, 35K08, 47D08 |
| url | https://arxiv.org/abs/2504.17879 |