Homogenisation On Homogeneous Spaces

TL;DR

This paper studies the limiting behavior of interpolated hypo-elliptic diffusions on homogeneous spaces as a parameter approaches zero, revealing new limit processes that are classified algebraically and geometrically, with convergence estimates.

Contribution

It introduces a family of interpolation equations on homogeneous spaces, analyzes their limits as the parameter tends to zero, and classifies the resulting processes using algebraic and geometric methods.

Findings

Convergence of stochastic processes on $G/H$ as $rac{1}{ ext{epsilon}}$ scale

Limit processes are not necessarily Brownian motions

Provides estimates on convergence rate in Wasserstein distance

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 $H$.