2018/04/01 by Xue-Mei Li, Xue-Mei LI
Mathematics · Medicine · #Advanced Neuroimaging Techniques and Applications #Brownian motion #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Invariant (physics) #Lie algebra #Lie group #Mathematical analysis #Mathematical physics #Mathematics #Pure mathematics #Scaling #Scaling limit
paper · pdf · doi:10.2969/jmsj/07027546
openalex created_date 2017/02/17 · crossref issued 2018/04/01 · crossref published 2018/04/01 · crossref published-print 2018/04/01 · openalex publication_date 2018/04/01 · crossref created 2018/04/18 · crossref deposited 2022/08/20 · crossref indexed 2026/07/31 · openalex updated_date 2026/08/01
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 Lε=(1/ε) ∑k (Ak)2+(1/ε) A0+Y0 where Y0, Ak are left invariant vector fields and \Ak\ generate the Lie-algebra of H.