A Natural Diffusion Distance and Equivalence of Local Convergence and Local Equicontinuity for a General Symmetric Diffusion Semigroup. (2nd October 2018)

• Record Type: Journal Article
• Authors: Goldberg, Maxim J.; Kim, Seonja
• Other Names: Ezzinbi Khalil Academic Editor.
• Is Part Of: Abstract and applied analysis. Volume 2018(2018)
• Publication Date: 2018-10-02
• Subjects: Mathematical analysis -- Periodicals; Mathematical analysis; Applied Mathematics; Mathematical Analysis; Periodicals; 515.05
• DOI: 10.1155/2018/6281504
• Languages: English
• ISSNs: 1085-3375

Abstract

Abstract : In this paper, we consider a general symmetric diffusion semigroupT t f t ≥ 0 on a topological spaceX with a positiveσ -finite measure, given, fort > 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 ofT t f tof is equivalent to local equicontinuity (int ) of the familyT t f t ≥ 0 . As a corollary of our main result, we show that, fort 0 > 0, T t + t 0 f converges locally toT t 0 f, ast converges to0 + . In the Appendix, we show that for very general metricsD onX, not necessarily arising from diffusion, ∫ X ‍ ρ t ( x, y ) D ( x, y ) d y → 0 a.e., ast → 0 + . R. Coifman and W. Leeb have assumed a quantitative version of this convergence, uniformly inx, in their recent work introducing a family of multiscale diffusion distances and establishing quantitative results about the equivalence of a bounded functionf being Lipschitz, and the rate of convergence ofT t f tof, ast → 0 + . We do not make such an assumption in the present work.