Heat kernel, eigenfunctions, and conditioned Brownian motion in planar domains (Q1122874)

Let \(\Omega\) be a planar domain of finite area, and consider the heat equation with Dirichlet boundary condition in \(\Omega\). Let \(\lambda\) and \(\phi\) be the first (positive) eigenvalue and -function, respectively. The main result is that - after a normalisation - for \(x\in \Omega\), \[ \lim_{t\to \infty}e^{\lambda t}p(t,x,y)/\phi (x)\phi (y)=1 \] uniformly in \(\Omega\), where p is the fundamental solution of the equation.