Imperial College London
Portrait Picture

Professor Xue-Mei Li

Faculty of Natural Sciences, Department of Mathematics

Chair in Probability and Stochastic Analysis
 
 
 
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Contact

 

+44 (0)20 7594 3709xue-mei.li Website CV

 
 
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Location

 

6M51Huxley BuildingSouth Kensington Campus

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Summary

 

Publications

Citation

BibTex format

@article{Li:2018:jmsj/07027546,
author = {Li, X-M},
doi = {jmsj/07027546},
journal = {Journal of Mathematical Society of Japan},
pages = {519--572},
title = {Homogenisation on homogeneous spaces},
url = {http://dx.doi.org/10.2969/jmsj/07027546},
volume = {70},
year = {2018}
}
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RIS format (EndNote, RefMan)

TY  - JOUR
AB  - 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 $H$.
AU  - Li,X-M
DO  - jmsj/07027546
EP  - 572
PY  - 2018///
SN  - 0025-5645
SP  - 519
TI  - Homogenisation on homogeneous spaces
T2  - Journal of Mathematical Society of Japan
UR  - http://dx.doi.org/10.2969/jmsj/07027546
UR  - https://projecteuclid.org/euclid.jmsj/1524038666#info
UR  - http://hdl.handle.net/10044/1/57321
VL  - 70
ER  -
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