Abstract and Applied Analysis

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

Maxim J. Goldberg and Seonja Kim

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In this paper, we consider a general symmetric diffusion semigroup T t f t 0 on a topological space X with a positive σ -finite measure, given, for t > 0 , by an integral kernel operator: T t f ( x ) X ρ t ( x , y ) f ( y ) d y . 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 T t f to f is equivalent to local equicontinuity (in t ) of the family T t f t 0 . As a corollary of our main result, we show that, for t 0 > 0 , T t + t 0 f converges locally to T t 0 f , as t converges to 0 + . In the Appendix, we show that for very general metrics D on X , not necessarily arising from diffusion, X ρ t ( x , y ) D ( x , y ) d y 0   a.e. , as t 0 + . R. Coifman and W. Leeb have assumed a quantitative version of this convergence, uniformly in x , in their recent work introducing a family of multiscale diffusion distances and establishing quantitative results about the equivalence of a bounded function f being Lipschitz, and the rate of convergence of T t f to f , as t 0 + . We do not make such an assumption in the present work.

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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

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