2021/09/08 by Wojciech Cygan, Cygan, Wojciech, Kamil Kaleta +3 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Spectral Theory in Mathematical Physics #advanced mathematical theories #math.FA #math.PR #math.SP #msc:05C81 #msc:31C05 #msc:35P05 #msc:39A70 #msc:47D08 #msc:60J10 #msc:60J75 #msc:81Q10 #quant-ph
paper · pdf · doi:10.48550/arxiv.2109.03788
28 pages, 1 figure
arxiv created 2021/09/08 · arxiv updated 2021/09/09
We propose and study a certain discrete time counterpart of the classical Feynman--Kac semigroup with a confining potential in countable infinite spaces. For a class of long range Markov chains which satisfy the direct step property we prove sharp estimates for functions which are (sub-, super-)harmonic in infinite sets with respect to the discrete Feynman--Kac operators. These results are compared with respective estimates for the case of a nearest-neighbour random walk which evolves on a graph of finite geometry. We also discuss applications to the decay rates of solutions to equations involving graph Laplacians and to eigenfunctions of the discrete Feynman--Kac operators. We include such examples as non-local discrete Schrödinger operators based on fractional powers of the nearest-neighbour Laplacians and related quasi-relativistic operators. Finally, we analyse various classes of Markov chains which enjoy the direct step property and illustrate the obtained results by examples.