Brownian potential theory and the Dirichlet problem
/ Dirichlet: DEE-ree-shleh /
There is a deep and exact correspondence between Brownian motion and classical potential theory: the partial differential equations of electrostatics and heat flow are solved by averaging over Brownian paths. The cornerstone is the Dirichlet problem — find a harmonic function inside a domain with prescribed boundary values — and Brownian motion solves it by a single, intuitive recipe: run the motion until it exits the domain, then read off the boundary value where it lands.
Precisely, let D be a bounded (nice) domain in R^d, let g be a continuous function on its boundary, and let tau_D = inf{ t : B_t not in D } be the first exit time. The function u(x) = E_x[ g(B_(tau_D)) ] — the expected boundary value seen by a Brownian motion started at x — is harmonic in D (it satisfies Laplace's equation Delta u = 0) and, under a mild boundary regularity condition, extends continuously to g on the boundary. This works because (1/2) Delta is the generator of Brownian motion, so u(B_t) is a local martingale precisely when u is harmonic (Dynkin's formula / Ito), and the mean-value property of harmonic functions IS the statement that Brownian motion started at x, stopped on a sphere, lands uniformly. More generally the Poisson equation (1/2) Delta u = -f with boundary data g has the Feynman-Kac-type solution u(x) = E_x[ g(B_(tau_D)) + integral over [0, tau_D] of f(B_s) ds ], and adding killing at rate c gives a Schrodinger-operator version. The harmonic measure — the law of the exit position B_(tau_D) started from x — is the kernel that represents every solution.
Why it matters: this is the original and still most powerful link between analysis and probability, giving probabilistic proofs of the maximum principle, Harnack inequalities, and boundary regularity (Wiener's criterion), and underpinning Monte Carlo solvers for elliptic PDE (walk-on-spheres). It also recasts physical capacity and equilibrium charge distributions as Brownian hitting probabilities. The honest caveats: the representation needs the boundary point to be REGULAR (Brownian motion started just inside actually exits near it immediately) — irregular points (the tip of a Lebesgue thorn) can fail, and Wiener's criterion characterises exactly which points are regular via capacity. In dimension d >= 2, single points are polar (never hit), so 'boundary values' on a too-small set are invisible to the process; and the clean (1/2) Delta generator means the constant 1/2 must be tracked or the Poisson term is off by a factor.
To find the equilibrium temperature at an interior point x of a metal plate with prescribed edge temperatures g, release many Brownian motions from x, let each diffuse until it hits the edge, and average the edge temperature at the landing points: that average is u(x), the solution of Laplace's equation.
Solve Delta u = 0 with boundary data g by averaging g over Brownian exit positions: u(x) = E_x[g(B at exit)].
The probabilistic solution needs boundary points to be regular (Wiener's criterion) — at irregular points the continuous extension can fail; and the generator is (1/2) Delta, so the constant 1/2 must be tracked in the Poisson/Feynman-Kac term.