## Abstract and Applied Analysis

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

#### Abstract

In this paper, we consider a general symmetric diffusion semigroup ${\{{T}_{t}f\}}_{t\ge \mathrm{0}}$ on a topological space $X$ with a positive $\sigma$-finite measure, given, for $t>\mathrm{0}$, by an integral kernel operator: ${T}_{t}f(x)\triangleq {\int }_{X}\mathrm{‍}{\rho }_{t}(x,y)f(y)dy$. 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 ${\{{T}_{t}f\}}_{t\ge \mathrm{0}}$. As a corollary of our main result, we show that, for ${t}_{\mathrm{0}}>\mathrm{0}$, ${T}_{t+{t}_{\mathrm{0}}}f$ converges locally to ${T}_{{t}_{\mathrm{0}}}f$, as $t$ converges to ${\mathrm{0}}^{+}$. In the Appendix, we show that for very general metrics $\mathcal{D}$ on $X$, not necessarily arising from diffusion, ${\int }_{X}\mathrm{‍}{\rho }_{t}(x,y)\mathcal{D}(x,y)dy\to \mathrm{0}\text{\hspace\{0.17em\}\hspace\{0.17em\}a.e.}$, as $t\to {\mathrm{0}}^{+}.$ 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 $t\to {\mathrm{0}}^{+}$. We do not make such an assumption in the present work.

#### Article information

Source
Abstr. Appl. Anal., Volume 2018 (2018), Article ID 6281504, 9 pages.

Dates
Accepted: 17 September 2018
First available in Project Euclid: 16 November 2018

https://projecteuclid.org/euclid.aaa/1542337408

Digital Object Identifier
doi:10.1155/2018/6281504

Mathematical Reviews number (MathSciNet)
MR3864591

Zentralblatt MATH identifier
07029291

#### Citation

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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