Two-sided bounds for degenerate processes with densities supported in subsets of R^N

TL;DR

This paper establishes precise two-sided bounds for the density of stochastic processes under weak Hörmander conditions, especially when the support is limited or varies with initial and final points, using PDE and Malliavin calculus techniques.

Contribution

It provides new two-sided bounds for densities of degenerate processes with restricted support, considering different asymptotic regimes based on initial and final conditions.

Findings

Derived lower bounds using Harnack inequalities for PDEs

Obtained upper bounds via probabilistic representation and Malliavin calculus

Analyzed cases with support not covering the entire space

Abstract

We obtain two-sided bounds for the density of stochastic processes satisfying a weak H\"ormander condition. In particular we consider the cases when the support of the density is not the whole space and when the density has various asymptotic regimes depending on the starting/final points considered (which are as well related to the number of brackets needed to span the space in H\"ormander's theorem). The proofs of our lower bounds are based on Harnack inequalities for positive solutions of PDEs whereas the upper bounds derive from the probabilistic representation of the density given by the Malliavin calculus.