A Natural Diffusion Distance and Equivalence of Local Convergence and Local Equicontinuity for a General Symmetric Diffusion Semigroup
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 specificatio...
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2018-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2018/6281504 |
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| author | Maxim J. Goldberg Seonja Kim |
| author_facet | Maxim J. Goldberg Seonja Kim |
| author_sort | Maxim J. Goldberg |
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| description | 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. |
| format | Article |
| id | doaj-art-5a95a7f7f95a447ea11e963486908833 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2018-01-01 |
| publisher | Wiley |
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| series | Abstract and Applied Analysis |
| spelling | doaj-art-5a95a7f7f95a447ea11e9634869088332025-08-20T03:34:58ZengWileyAbstract and Applied Analysis1085-33751687-04092018-01-01201810.1155/2018/62815046281504A Natural Diffusion Distance and Equivalence of Local Convergence and Local Equicontinuity for a General Symmetric Diffusion SemigroupMaxim J. Goldberg0Seonja Kim1Theoretical and Applied Sci., Ramapo College of NJ, Mahwah, NJ 07430, USAMathematics Dept., Middlesex County College, Edison, NJ 08818, USAIn 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.http://dx.doi.org/10.1155/2018/6281504 |
| spellingShingle | Maxim J. Goldberg Seonja Kim A Natural Diffusion Distance and Equivalence of Local Convergence and Local Equicontinuity for a General Symmetric Diffusion Semigroup Abstract and Applied Analysis |
| title | A Natural Diffusion Distance and Equivalence of Local Convergence and Local Equicontinuity for a General Symmetric Diffusion Semigroup |
| title_full | A Natural Diffusion Distance and Equivalence of Local Convergence and Local Equicontinuity for a General Symmetric Diffusion Semigroup |
| title_fullStr | A Natural Diffusion Distance and Equivalence of Local Convergence and Local Equicontinuity for a General Symmetric Diffusion Semigroup |
| title_full_unstemmed | A Natural Diffusion Distance and Equivalence of Local Convergence and Local Equicontinuity for a General Symmetric Diffusion Semigroup |
| title_short | A Natural Diffusion Distance and Equivalence of Local Convergence and Local Equicontinuity for a General Symmetric Diffusion Semigroup |
| title_sort | natural diffusion distance and equivalence of local convergence and local equicontinuity for a general symmetric diffusion semigroup |
| url | http://dx.doi.org/10.1155/2018/6281504 |
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