Solution of the Dirichlet problem for the equation $a\Delta u+b\cdot \nabla u=0$ by the Monte Carlo method

TL;DR

This paper develops a Monte Carlo method to solve the Dirichlet problem for a linear PDE with drift, using probabilistic representations and mean value properties to establish existence, uniqueness, and explicit solutions.

Contribution

It introduces a probabilistic approach using drifted Brownian motion to explicitly solve the Dirichlet problem for a linear PDE with drift, including convergence and boundary regularity conditions.

Findings

Established a mean value property equivalent to the PDE

Constructed a convergent family of random variables for solution representation

Proved uniqueness and existence under boundary regularity

Abstract

In this paper we study the Dirichlet problem corresponding to an open bounded set $D\subset \mathbb{R}^{d}$ and the operator \begin{equation*} A=\sum_{i=1}^{d}a\frac{\partial ^{2}}{\partial x_{i}^{2}} +\sum_{i=1}^{d}b_{i}\frac{\partial }{\partial x_{i}}, \end{equation*} where $a>0$ and $b\in \mathbb{R}^{d}$. We define a mean value property and prove that a function $u$ has such property in $D$ if and only if $Au=0$ in $D$. Using this characterization, and a drifted Brownian motion, we define a family of random variables that converges almost surely and the limit is used to give an explicit representation for the solutions to the Dirichlet problem, this immediately implies the uniqueness. On the other hand, the existence of the solution is proved imposing a regular condition on the boundary of $D$.