Cauchy-Dirichlet problems for a class of hypoelliptic equation in $\R^d$: q new probabilistic representation formula for the gradient of the solutions

TL;DR

This paper establishes the existence of solutions and introduces a novel probabilistic formula for the gradient of the semigroup related to an Ornstein-Uhlenbeck process within bounded convex domains in multiple dimensions.

Contribution

It provides a new probabilistic representation formula for the gradient of solutions to a class of hypoelliptic equations in bounded convex domains.

Findings

Existence of solutions to the hypoelliptic PDEs.

A new probabilistic formula for the gradient of the semigroup.

Application to Ornstein-Uhlenbeck processes in convex domains.

Abstract

We prove the existence and a new representation formula for the gradient of the semigroup associated to an Ornstein-Uhlenbeck in a bounded convex domain in d dimensions.