## 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

Maxim J. Goldberg and Seonja Kim

#### Abstract

In this paper, we consider a general symmetric diffusion semigroup ${\left\{{T}_{t}f\right\}}_{t\ge \mathrm{0}}$ on a topological space $X$ with a positive $\sigma $-finite measure, given, for $t>\mathrm{0}$, by an integral kernel operator: ${T}_{t}f(x)\triangleq {\int}_{X}\mathrm{\u200d}{\rho}_{t}(x,y)f(y)dy$. 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 ${\left\{{T}_{t}f\right\}}_{t\ge \mathrm{0}}$. 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, ${\int}_{X}\mathrm{\u200d}{\rho}_{t}(x,y)\mathcal{D}(x,y)dy\to \mathrm{0}\text{\hspace{0.17em}\hspace{0.17em}a.e.}$, 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**

Received: 27 July 2018

Accepted: 17 September 2018

First available in Project Euclid: 16 November 2018

**Permanent link to this document**

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