Open Access
April, 2018 Homogenisation on homogeneous spaces
Xue-Mei LI
J. Math. Soc. Japan 70(2): 519-572 (April, 2018). DOI: 10.2969/jmsj/07027546


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 $H$.


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Xue-Mei LI. "Homogenisation on homogeneous spaces." J. Math. Soc. Japan 70 (2) 519 - 572, April, 2018.


Published: April, 2018
First available in Project Euclid: 18 April 2018

zbMATH: 06902434
MathSciNet: MR3787732
Digital Object Identifier: 10.2969/jmsj/07027546

Primary: 58J65 , 58J70 , 60GXX , 60Hxx

Keywords: adiabatic limit , classification of effective dynamics , diffusion creation , Hörmander’s conditions , stochastic averaging

Rights: Copyright © 2018 Mathematical Society of Japan

Vol.70 • No. 2 • April, 2018
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