Homogenisation On Homogeneous Spaces

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 $\frac{1}{\epsilon }$), 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 ${\mathcal{L}}^{\epsilon }=\frac{1}{\epsilon }{\sum }_k(A_k{)}^2+\frac{1}{\epsilon }{A}_0+Y_0$ where $Y_0,A_k$ are left invariant vector fields and $\{A_k\}$ generate the Lie-algebra of $H$.

52 pages, to appear: Journal of the Mathematical Society of Japan. With an appendix by D. Rumynin

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Cited by in corpus (3)

• Insight into High-order Harmonic Generation from Solids: The Contributions of the Bloch Wave-packets Moving on the Group and Phase Velocities
• Orientation-dependent transition rule in high-order harmonic generation from solids
• High-order harmonic generation from 2D periodic potentials in circularly and bichromatic circularly polarized laser fields