Homogenisation on homogeneous spaces

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Title: Homogenisation on homogeneous spaces
Authors: Hairer, X
Item Type: Journal Article
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 > 0 , interpolating hypo- elliptic diffusions on H and translates of exponential maps on G . Assume that G/H is a reductive homogeneous space, in the sense of Nomizu. We construct a family of stochastic processes on G (whose projections to the coset space G/H agree with that of the solutions) that converge as → 0 . Over the time scale 1 / , they converge to a Markov process and the latter 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 us to conclude whether their projections to the homogeneous space G/H are Markov processes. From the point of view of operators, this can be considered as “taking the adiabatic limit” of L = 1 ∑ k ( A k ) 2 + 1 A 0 + Y 0 where Y 0 ,A k are left invariant vector fields and { A k } generate the Lie-algebra of H . Furthermore an upper bound for the rate of the convergence in the Wasserstein distance is given, and we also conclude that the parallel translations along the diffusions converge to stochastic parallel translations along the limit diffusions.
Issue Date: 31-Dec-2018
Date of Acceptance: 3-Feb-2017
URI: http://hdl.handle.net/10044/1/57321
ISSN: 0025-5645
Publisher: Mathematical Society of Japan
Journal / Book Title: Journal of Mathematical Society of Japan
Copyright Statement: This paper is embargoed until publication.
Keywords: 0101 Pure Mathematics
General Mathematics
Publication Status: Accepted
Embargo Date: publication subject to indefinite embargo
Appears in Collections:Pure Mathematics
Mathematics
Faculty of Natural Sciences



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