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 𝜀>0, interpolating hypo-elliptic diffusions on 𝐻 and translates of exponential maps on 𝐺 and examine the dynamics as 𝜀→ 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/𝜀), 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 ℒ^𝜀=(1/𝜀) ∑_𝑘 (𝐴_𝑘)²+(1/𝜀) 𝐴₀+𝑌₀ where 𝑌₀, 𝐴_𝑘 are left invariant vector fields and {𝐴_𝑘} generate the Lie-algebra of 𝐻.