The Dirichlet Problem

Steady-state temperature from a random walker: u(x) is the average boundary value where Brownian motion exits.

The idea

The Dirichlet problem asks for a function inside a domain from its values on the edge: given a domain $D$ and boundary values $f$ on its boundary $\partial D$, find the function $u$ on $D$ that equals $f$ on the boundary and is in steady state inside. The model to keep in mind is the steady temperature of a metal plate whose edge is held at the profile $f$: once heat stops flowing, the interior is determined by the edge alone.

Steady state is an averaging condition. Nothing changes at an interior point exactly when its temperature equals the average of the temperatures around it, and a function with this property is called harmonic. The average has a probabilistic description.

Theorem (Brownian solution of the Dirichlet problem).

Let $u$ be the solution of the Dirichlet problem on $D$ with boundary values $f$. For $x \in D$, release a Brownian particle at $x$ and let $\tau$ be the first time it leaves $D$. Then $u(x) = \mathbb{E}_{x}\big[f(B_{\tau})\big].$

The steady temperature at $x$ is the mean boundary temperature at the exit point of a Brownian particle released at $x$. The distribution of that exit point on the boundary is called harmonic measure; a point near a hot stretch of edge exits through it often, which is why it comes out hot.

The same statement holds on a graph, where harmonic means the average over the neighbours and the particle is a random walk.

Ways to work on it

• Walkthrough. Steady state on a graph, the exit-average formula u(x) = E_x[f(B_ )], and harmonic measure.
• Proof. The mean value property from rotational invariance plus the strong Markov property.
• Practice. Centre-of-disc averages against uniform harmonic measure, and two-site discrete Dirichlet systems.
• Hardest. A square plate: symmetry supplies harmonic measure at the centre where no formula does.

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