Heat potentials, thermal capacity and brownian motion


Per-Anders Ivert


The Perron method for solving the Dirichlet problem (first
boundary-value problem) in a bounded domain in $\mathbf{R}^{d}$ can be
used also for the heat equation in a bounded domain in
$\mathbf{R}^{d}\times \mathbf{R}$, and it yields a solution operator,
assigning to each continuous boundary value function a solution of the
equation. The question of regularity of a boundary point (i.e. a point
where the solution assumes the prescribed value) arises, and it is
equivalent to the question of regularity for the heat potential of a
compact set. We discuss the generalization of the famous Wiener
criterion for such regularity. The value of the heat potential at a
point of time-space can be interpreted either as a temperature or as
the probability of hitting the compact set with a Brownian motion
starting at that point.