Stopping times and $\Gamma$-convergence

Abstract:

The equation $\partial u/\partial t = \Delta u - \mu u$ represents diffusion with killing. The strength of the killing is described by the measure $\mu$, which is not assumed to be finite or even $\sigma$-finite (to illustrate the effect of infinite values for $\mu$, it may be noted that the diffusion is completely absorbed on any set $A$ such that $\mu (B) = \infty$ for every nonpolar subset $B$ of $A$). In order to give rigorous mathematical meaning to this general diffusion equation with killing, one may interpret the solution $u$ as arising from a variational problem, via the resolvent, or one may construct a semigroup probabilistically, using a multiplicative functional. Both constructions are carried out here, shown to be consistent, and applied to the study of the diffusion equation, as well as to the study of the related Dirichlet problem for the equation $\Delta u - \mu u = 0$. The class of diffusions studied here is closed with respect to limits when the domain is allowed to vary. Two appropriate forms of convergence are considered, the first being $\gamma$-convergence of the measures $\mu$, which is defined in terms of the variational problem, and the second being stable convergence in distribution of the multiplicative functionals associated with the measures $\mu$. These two forms of convergence are shown to be equivalent.