= Kakutani solution of the Dirichlet problem
{c}
{title2=$u(x)=\mathbb E_x[f(B_{\tau_D})]$}
For a bounded <domain> whose boundary points are all <regular boundary points>, continuous <Dirichlet boundary data> $f$ have a unique solution
$$
u(x)=\mathbb E_x[f(B_{\tau_D})],\qquad\Delta u=0\text{ in }D,\quad u|_{\partial D}=f.
$$
Here $\tau_D$ is the <Brownian exit time>. The <Strong Markov property> gives the <mean value property for harmonic functions>; a <barrier for the Dirichlet problem> gives continuity at the boundary. This is an existence theorem as well as the <Brownian representation of the Dirichlet problem> for an already known solution.
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