Harnack inequalities and Bounds for Densities of Stochastic Processes

TL;DR

This paper develops methods using Harnack inequalities to establish lower bounds for the densities of stochastic processes associated with degenerate parabolic operators satisfying Hörmander's condition, with applications to various examples.

Contribution

It introduces a novel approach based on Harnack chains to derive lower bounds for fundamental solutions of degenerate parabolic PDEs linked to stochastic processes.

Findings

Established lower bounds for fundamental solutions using Harnack inequalities.

Discussed PDE and SDE techniques for upper bounds.

Provided examples of operators where the method applies.

Abstract

We consider possibly degenerate parabolic operators in the form $$ \sum_{k=1}^{m}X_{k}^{2}+X_{0}-\partial_{t}, $$ that are naturally associated to a suitable family of stochastic differential equations, and satisfying the H\"ormander condition. Note that, under this assumption, the operators in the form $\L$ has a smooth fundamental solution that agrees with the density of the corresponding stochastic process. We describe a method based on Harnack inequalities and on the construction of Harnack chains to prove lower bounds for the fundamental solution. We also briefly discuss PDE and SDE methods to prove analogous upper bounds. We eventually give a list of meaningful examples of operators to which the method applies.