A Natural Diffusion Distance and Equivalence of Local Convergence and Local Equicontinuity for a General Symmetric Diffusion Semigroup
Joint Authors
Source
Issue
Vol. 2018, Issue 2018 (31 Dec. 2018), pp.1-9, 9 p.
Publisher
Hindawi Publishing Corporation
Publication Date
2018-10-02
Country of Publication
Egypt
No. of Pages
9
Main Subjects
Abstract EN
In this paper, we consider a general symmetric diffusion semigroup Ttft≥0 on a topological space X with a positive σ-finite measure, given, for t>0, by an integral kernel operator: Ttf(x)≜∫Xρt(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 Ttf to f is equivalent to local equicontinuity (in t) of the family Ttft≥0.
As a corollary of our main result, we show that, for t0>0, Tt+t0f converges locally to Tt0f, as t converges to 0+.
In the Appendix, we show that for very general metrics D on X, not necessarily arising from diffusion, ∫Xρt(x,y)D(x,y)dy→0 a.e., as t→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 Ttf to f, as t→0+.
We do not make such an assumption in the present work.
American Psychological Association (APA)
Goldberg, Maxim J.& Kim, Seonja. 2018. A Natural Diffusion Distance and Equivalence of Local Convergence and Local Equicontinuity for a General Symmetric Diffusion Semigroup. Abstract and Applied Analysis،Vol. 2018, no. 2018, pp.1-9.
https://search.emarefa.net/detail/BIM-1114288
Modern Language Association (MLA)
Goldberg, Maxim J.& Kim, Seonja. A Natural Diffusion Distance and Equivalence of Local Convergence and Local Equicontinuity for a General Symmetric Diffusion Semigroup. Abstract and Applied Analysis No. 2018 (2018), pp.1-9.
https://search.emarefa.net/detail/BIM-1114288
American Medical Association (AMA)
Goldberg, Maxim J.& Kim, Seonja. A Natural Diffusion Distance and Equivalence of Local Convergence and Local Equicontinuity for a General Symmetric Diffusion Semigroup. Abstract and Applied Analysis. 2018. Vol. 2018, no. 2018, pp.1-9.
https://search.emarefa.net/detail/BIM-1114288
Data Type
Journal Articles
Language
English
Notes
Includes bibliographical references
Record ID
BIM-1114288