A Natural Diffusion Distance and Equivalence of Local Convergence and Local Equicontinuity for a General Symmetric Diffusion Semigroup

In this paper, we consider a general symmetric diffusion semigroup on a topological space with a positive -finite measure, given, for , by an integral kernel operator: . As one of the contributions of our paper, we define a diffusion distance whose specification follows naturally from imposing a reasonable Lipschitz condition on diffused versions of arbitrary bounded functions. We next show that the mild assumption we make, that balls of positive radius have positive measure, is equivalent to a similar, and an even milder looking, geometric demand. In the main part of the paper, we establish that local convergence of to is equivalent to local equicontinuity (in ) of the family . As a corollary of our main result, we show that, for , converges locally to , as converges to . In the Appendix, we show that for very general metrics on , not necessarily arising from diffusion, , as R. Coifman and W. Leeb have assumed a quantitative version of this convergence, uniformly in , in their recent work introducing a family of multiscale diffusion distances and establishing quantitative results about the equivalence of a bounded function being Lipschitz, and the rate of convergence of to , as . We do not make such an assumption in the present work.