  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.
