Abstract and Applied Analysis
- Abstr. Appl. Anal.
- Volume 2018 (2018), Article ID 6281504, 9 pages.
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.
Abstr. Appl. Anal., Volume 2018 (2018), Article ID 6281504, 9 pages.
Received: 27 July 2018
Accepted: 17 September 2018
First available in Project Euclid: 16 November 2018
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Goldberg, Maxim J.; Kim, Seonja. A Natural Diffusion Distance and Equivalence of Local Convergence and Local Equicontinuity for a General Symmetric Diffusion Semigroup. Abstr. Appl. Anal. 2018 (2018), Article ID 6281504, 9 pages. doi:10.1155/2018/6281504. https://projecteuclid.org/euclid.aaa/1542337408