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

Abstract We develop a parametrix approach for constructing solutions and establishing grid-size-independent estimates for semidiscrete 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 semidiscrete 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.