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 ${\left\{T_{t}f\right\}}_{t\ge \mathrm{0}}$ on a topological space $X$ with a positive $\sigma$-finite measure, given, for $t>\mathrm{0}$, by an integral kernel operator: $T_{t}f(x)\triangleq {\int }_X\mathrm{‍}{\rho }_t(x,y)f(y)dy$. 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, ${\int }_X\mathrm{‍}{\rho }_t(x,y)\mathcal{D}(x,y)dy\to \mathrm{0}\text{a.e.}$, 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.