Decay of harmonic functions for discrete time Feynman--Kac operators with confining potentials

TL;DR

This paper investigates the decay properties of harmonic functions related to discrete Feynman--Kac operators with confining potentials on infinite graphs, providing sharp estimates and applications to non-local Schrödinger operators.

Contribution

It introduces a discrete-time Feynman--Kac framework for infinite graphs, deriving sharp harmonic function estimates and exploring applications to non-local operators and Markov chains.

Findings

Sharp estimates for harmonic functions on infinite graphs.

Comparison between long-range and nearest-neighbor cases.

Applications to decay rates of solutions and eigenfunctions.

Abstract

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 Schr\"odinger 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.