This result connects Brownian motion to classical PDE theory: the function $u(x) = \mathbb{E}_x[\varphi(B_\tau)]$, defined by averaging the boundary data $\varphi$ over the exit distribution of Brownian motion from a bounded domain $D$, is harmonic on $D$. This is the probabilistic solution of the Dirichlet problem — the problem of finding a harmonic function with prescribed boundary values — and it provides a powerful alternative to the classical analytical methods (Perron's method, integral equation methods, variational methods).

The proof establishes harmonicity by verifying the mean value property. For any ball $B(x, \delta) \subset D$, the [Strong Markov Property](/theorems/1180) applied at the first exit time $\tau'$ from the ball gives $u(x) = \mathbb{E}_x[u(B_{\tau'})]$. The rotational invariance of Brownian motion ensures that $B_{\tau'}$ is uniformly distributed on the sphere $\partial B(x, \delta)$, so $u(x)$ equals the spherical average of $u$ over $\partial B(x, \delta)$. This is the mean value property, which characterises harmonic functions.

The probabilistic approach to the Dirichlet problem has two significant advantages. First, it provides an explicit representation formula for the solution, expressing it as an expectation that can in principle be computed or estimated by Monte Carlo simulation. Second, the probabilistic framework clarifies the role of domain geometry: the boundary behaviour of the solution (whether $u$ extends continuously to $\partial D$ with $u = \varphi$ on the boundary) depends on the probability that Brownian motion exits through each boundary point, which is controlled by geometric conditions such as the [Poincare Cone Condition](/theorems/1188). Uniqueness of the Dirichlet solution, given continuity up to the boundary, is established separately via the [Maximum Principle](/theorems/1187).