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Homogenisation on homogeneous spaces
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Li, X-M. (2018) Homogenisation on homogeneous spaces. Journal of the Mathematical Society of Japan, 70 (2). pp. 519-572. doi:10.2969/jmsj/07027546
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Official URL: http://doi.org/10.2969/jmsj/07027546
Abstract
Motivated by collapsing of Riemannian manifolds and inhomogeneous scaling of left invariant Riemannian metrics on a real Lie group G with a sub-group H, we consider a family of stochastic differential equations (SDEs) on G with parameter ϵ>0 and Markov generator Lϵ=1ϵ∑k(Ak)2+1ϵA0+Y0 where Y0,Ak are left invariant vector fields and {Ak} generate the Lie-algebra of H. Assuming that G/H is a reductive homogeneous space, in the sense of Nomizu, we study the solutions of the SDE as ϵ approaches zero.
We use the projection as a `conservation law' and obtain a family of slow variables: these are random variables, parameterized by ϵ, evolving in time. We examine their convergence on the time interval [0,1] and on [0,1ϵ], and study their effective motions. The effective motions on G over the larger time scale are Markov processes, which can be classified algebraically by a real Peter Weyl theorem and geometrically using a weak notion of the naturally reductive property; the classifications allow us to conclude when their projections to the homogeneous space G/H are Markov.
| Item Type: | Journal Article | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| Subjects: | Q Science > QA Mathematics | ||||||||
| Divisions: | Faculty of Science > Mathematics | ||||||||
| Library of Congress Subject Headings (LCSH): | Homogenization (Differential equations), Homogeneous spaces | ||||||||
| Journal or Publication Title: | Journal of the Mathematical Society of Japan | ||||||||
| Publisher: | Maths Society Japan | ||||||||
| ISSN: | 0025-5645 | ||||||||
| Official Date: | 18 April 2018 | ||||||||
| Dates: |
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| Date of first compliant deposit: | 8 March 2017 | ||||||||
| Volume: | 70 | ||||||||
| Number: | 2 | ||||||||
| Page Range: | pp. 519-572 | ||||||||
| DOI: | 10.2969/jmsj/07027546 | ||||||||
| Status: | Peer Reviewed | ||||||||
| Publication Status: | Published | ||||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||||
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