Journal of the Mathematical Society of Japan

Homogenisation on homogeneous spaces

Xue-Mei LI

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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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J. Math. Soc. Japan, Volume 70, Number 2 (2018), 519-572.

First available in Project Euclid: 18 April 2018

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Primary: 60Gxx: Stochastic processes 60Hxx: Stochastic analysis [See also 58J65] 58J65: Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60] 58J70: Invariance and symmetry properties [See also 35A30]

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


LI, Xue-Mei. Homogenisation on homogeneous spaces. J. Math. Soc. Japan 70 (2018), no. 2, 519--572. doi:10.2969/jmsj/07027546.

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