Semi-discrete heat equations with variable coefficients and the parametrix method

TL;DR

This paper introduces a parametrix method for semi-discrete heat equations with variable coefficients, deriving Lorentzian estimates to handle convolutions and establish solution existence and stability.

Contribution

It develops a novel parametrix approach for semi-discrete heat equations, overcoming the lack of Gaussian estimates by using Lorentzian bounds for variable coefficient cases.

Findings

Established grid-size independent estimates for solutions.

Proved convergence of the parametrix series in the semi-discrete setting.

Extended classical heat kernel analysis to variable coefficients with Lorentzian estimates.

Abstract

We develop a parametrix approach for constructing solutions and establishing grid-size independent estimates for semi-discrete heat equations with variable coefficients. While the classical continuous setting benefits from Gaussian estimates of the constant coefficient heat kernel, such estimates are not available in the semi-discrete context. To address this complication, we derive estimates involving products of heavy-tailed Lorentz (also known as Cauchy) probability densities. These Lorentzian estimates provide a sufficient handle on certain iterated convolutions involving Bessel functions, enabling us to achieve convergence of the parametrix approach.