Solving the Poisson equation using a random walk on $\mathbb Z ^d$

Question

How do I solve the Poisson equation with the help of a discrete random walk on $\mathbb Z ^d$?

Answer 1

Random walk method for the two‐ and three‐dimensional Laplace, Poisson and Helmholtz's equations (paywall)

Random Walk Method for Potential Problems (freely accessible)

The random walk method is developed for solving the Laplace, Poisson, and Helmholtz equations in two and three dimensions. Sin the random walk method is a local method, the solution at an arbitrary point can be determined without having to obtain the complete field solution. The method is based on the properties of diffusion processes, the Itô formula, the Dynkin formula, the Feynman–Kac functional, and Monte Carlo simulation.

Answer 2

If $\varphi(\cdot)$ is a function defined on a finite $\Omega \subset \mathbb{Z}^d$, define $$ f(x):=\mathbb{E}^x\left(\sum_{t=0}^{\tau-1}\varphi(X_t)\right), $$ where $\mathbb{E}^x$ means the expectation for the random walk started from $x$, and $\tau:=\min\{t:X_t\notin \Omega\}$. Then, by conditioning on the first step, you see that $f$ satisfies the equation $-\Delta f = \varphi$ and $f\equiv 0$ outside $\Omega$.

For $\Omega=\mathbb{Z}^d$, you can outright take $\tau=\infty$ if $d\geq 3$ (the expectation will be finite under reasonable assumptions on $\varphi$). For $d=2$, the walk is recurrent and you get infinity even with a finitely supported $\varphi$. One way to regularize is to put $$ f(x):=\lim_{n\to\infty} \left(\mathbb{E}^x\left(\sum_{t=0}^n\varphi(X_t)\right)-\mathbb{E}^0\left(\sum_{t=0}^n\varphi(X_t)\right)\right). $$ You can use e.g. a coupling of walks started from $0$ and $x$ to see that the limit exists.