Stochastic Homogenisation On Homogeneous Spaces

Xue-Mei Li

Published 2015-05-25Version 1

Let G be a Lie group with a closed subgroup H and a left invariant Riemannian metric. We study a family of stochastic differential equations arising from inhomogeneous scaling of the Riemannian metric. Such equations interpolate between horizontal exponential maps on G and diffusions on H. Suppose that the homogeneous space has a reductive decomposition and the horizontal direction is not invariant under the adjoint action of H. We obtain a pair of effective stochastic processes on G and on the homogeneous manifold M respectively and we classify their limits.