Weak limits of stopped diffusions

Abstract:

We consider the following homogenization problem: Let Brownian motion in ${{\mathbf {R}}^d}$, $d \geqslant 3$, be killed on the surface of many small absorbing bodies (standard diffusion equation with Dirichlet boundary conditions). We investigate the limit as the number of bodies approaches infinity and the size of the bodies approaches $0$. By taking a weak limit of stopping times we replace a convergence problem on the state space by an identification of the limit on the sample space. This technique then gives results without smoothness assumptions which were previously necessary.