Nash estimates and upper bounds for non-homogeneous Kolmogorov equations

TL;DR

This paper establishes Gaussian upper bounds for fundamental solutions of certain non-homogeneous Kolmogorov equations, extending classical results and applicable in stochastic processes, physics, and finance.

Contribution

It provides a generalized Gaussian upper bound for ultra-parabolic equations with non-smooth coefficients, broadening the scope of previous classical results.

Findings

Gaussian upper bounds are independent of coefficient smoothness

Generalizes Nash, Aronson, and Davies classical results

Applicable to stochastic processes, physics, and finance

Abstract

We prove a Gaussian upper bound for the fundamental solutions of a class of ultra-parabolic equations in divergence form. The bound is independent on the smoothness of the coefficients and generalizes some classical results by Nash, Aronson and Davies. The class considered has relevant applications in the theory of stochastic processes, in physics and in mathematical finance.