Asymptotics of parabolic Green's functions on lattices

TL;DR

This paper derives detailed asymptotic expansions for Green's functions of discrete parabolic equations on lattices, providing uniform estimates and extending results to higher-order approximations relevant for random walks and lattice dynamics.

Contribution

It introduces a method to obtain arbitrary-order asymptotic expansions of lattice Green's functions with uniform remainder estimates, generalizing to higher-order difference operators.

Findings

Derived asymptotic expansions up to arbitrary order

Provided uniform estimates of remainders across the lattice

Extended results to higher-order transition probabilities for random walks

Abstract

For parabolic spatially discrete equations, we consider Green's functions, also known as heat kernels on lattices. We obtain their asymptotic expansions with respect to powers of time variable $t$ up to an arbitrary order and estimate the remainders uniformly on the whole lattice. The spatially discrete (difference) operators under consideration are finite-difference approximations of continuous strongly elliptic differential operators (with constant coefficients) of arbitrary even order in $\mathbb R^d$ with arbitrary $d\in\mathbb N$. This genericity, besides numerical and deterministic lattice-dynamics applications, allows one to obtain higher-order asymptotics of transition probability functions for continuous-time random walks on $\mathbb Z^d$ and other lattices.