Homogenisation on homogeneous spaces

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Title: Homogenisation on homogeneous spaces
Authors: Li, X-M
Item Type: Journal Article
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 introduce a family of interpolation equations on $G$ with a parameter $\epsilon>0$, interpolating hypo-elliptic diffusions on $H$ and translates of exponential maps on $G$ and examine the dynamics as $\epsilon\to 0$. When $H$ is compact, we use the reductive homogeneous structure of Nomizu to extract a converging family of stochastic processes (converging on the time scale $1/\epsilon$), proving the convergence of the stochastic dynamics on the orbit spaces $G/H$ and their parallel translations, providing also an estimate on the rate of the convergence in the Wasserstein distance. Their limits are not necessarily Brownian motions and are classified algebraically by a Peter-Weyl's theorem for real Lie groups and geometrically using a weak notion of the naturally reductive property; the classifications allow to conclude the Markov property of the limit process. This can be considered as "taking the adiabatic limit" of the differential operators $\mathcal{L}^\epsilon=(1/\epsilon) \sum_k (A_k)^2+(1/\epsilon) A_0+Y_0$ where $Y_0, A_k$ are left invariant vector fields and $\{A_k\}$ generate the Lie-algebra of $H$.
Issue Date: 2018
Date of Acceptance: 3-Feb-2017
URI: http://hdl.handle.net/10044/1/57321
DOI: https://doi.org/10.2969/jmsj/07027546
ISSN: 0025-5645
Publisher: Mathematical Society of Japan
Start Page: 519
End Page: 572
Journal / Book Title: Journal of Mathematical Society of Japan
Volume: 70
Issue: 2
Copyright Statement: © Mathematical Society of Japan. Reuse is permitted for education, research and other academic purposes
Keywords: General Mathematics
0101 Pure Mathematics
Publication Status: Published
Online Publication Date: 2018-04-18
Appears in Collections:Pure Mathematics
Faculty of Natural Sciences

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