Applebaum, D.B. and Le Ngan, T. (2018) Transition densities and traces for invariant feller processes on compact symmetric spaces. Potential Analysis, 49 (4). pp. 479-501. ISSN 0926-2601
Abstract
We find necessary and sufficient conditions for a finite K–bi–invariant measure on a compact Gelfand pair (G, K) to have a square–integrable density. For convolution semigroups, this is equivalent to having a continuous density in positive time. When (G, K) is a compact Riemannian symmetric pair, we study the induced transition density for G–invariant Feller processes on the symmetric space X = G/K. These are obtained as projections of K–bi–invariant L´evy processes on G, whose laws form a convolution semigroup. We obtain a Fourier series expansion for the density, in terms of spherical functions, where the spectrum is described by Gangolli’s L´evy–Khintchine formula. The density of returns to any given point on X is given by the trace of the transition semigroup, and for subordinated Brownian motion, we can calculate the short time asymptotics of this quantity using recent work of Ba˜nuelos and Baudoin. In the case of the sphere, there is an interesting connection with the Funk–Hecke theorem.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2017 The Author(s). This article is an open access publication. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. |
Keywords: | Feller process; Levy process; Lie group; Symmetric space; Convolution semigroup; Transition kernel; Spherical function; Trace; Subordination; Funk-Hecke theorem |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield) |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 01 Nov 2017 15:32 |
Last Modified: | 11 Jan 2024 12:59 |
Status: | Published |
Publisher: | Springer Verlag |
Refereed: | Yes |
Identification Number: | 10.1007/s11118-017-9664-4 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:123259 |
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