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

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.