Large time behaviour for the heat equation on $\Z,$ moments and decay rates

TL;DR

This paper investigates the large time decay and behavior of solutions to the discrete heat equation on the integer lattice, analyzing moments, decay rates, and asymptotics to understand how solutions evolve over time.

Contribution

It provides a comprehensive analysis of the moments and decay rates of the discrete heat equation, including asymptotic behaviors and optimal decay estimates using Fourier techniques.

Findings

Established asymptotic pointwise decay rates.

Proved mass conservation and moments properties.

Derived optimal decay rates for solutions in  spaces.

Abstract

The paper is devoted to understand the large time behaviour and decay of the solution of the discrete heat equation in the one dimensional mesh $\Z$ on $\ell^p$ spaces, and its analogies with the continuous-space case. We do a deep study of the moments of the discrete gaussian kernel (which is given in terms of Bessel functions), in particular the mass conservation principle; that is reflected on the large time behaviour of solutions. We prove asymptotic pointwise and $\ell^p$ decay results for the fundamental solution. We use that estimates to get rates on the $\ell^p$ decay and large time behaviour of solutions. For the $\ell^2$ case, we get optimal decay by use of Fourier techniques.