A natural diffusion distance and equivalence of local convergence and local equicontinuity for a general symmetric diffusion semigroup (Q1728532)

Summary: In this paper, we consider a general symmetric diffusion semigroup \(\left\{T_t f\right\}_{t \geq 0}\) on a topological space \(X\) with a positive \(\sigma\)-finite measure, given, for \(t > 0\), by an integral kernel operator: \(T_t f(x) \triangleq \int_X \rho_t(x, y) f(y) d y\). 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 \(\left\{T_t f\right\}_{t \geq 0}\). 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 \(\mathcal{D}\) on \(X\), not necessarily arising from diffusion, \(\int_X \rho_t(x, y) \mathcal{D}(x, y) d y \rightarrow 0\) a.e., as \(t \rightarrow 0^+ \). 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 \(t \rightarrow 0^+\). We do not make such an assumption in the present work.