Asymptotics of the heat kernels on 2D lattices

TL;DR

This paper derives detailed asymptotic expansions for heat kernels on 2D lattices, including a novel time-independent term, which are essential for analyzing complex spatio-temporal patterns in reaction-diffusion systems.

Contribution

It provides the first comprehensive asymptotic expansion of 2D lattice heat kernels, including a new time-independent component, with uniform estimates across the entire lattice.

Findings

Asymptotic expansions include a time-independent term unlike in 1D.

Uniform remainder estimates are established for the entire lattice.

Results are applicable to reaction-diffusion pattern analysis.

Abstract

We obtain asymptotic expansions of the spatially discrete 2D heat kernels, or Green's functions on lattices, with respect to powers of time variable up to an arbitrary order and estimate the remainders uniformly on the whole lattice. Unlike in the 1D case, the asymptotics contains a time independent term. The derivation of its spatial asymptotics is the technical core of the paper. Besides numerical applications, the obtained results play a crucial role in the analysis of spatio-temporal patterns for reaction-diffusion equations on lattices, in particular rattling patterns for hysteretic diffusion systems.