Title: Stochastic Homogenisation On Homogeneous Spaces
Speaker: Xue-Mei Li
Speaker Info: University of Warwick
Motivated by collapsing of Berger’s spheres to a lower dimensional manifold, we propose to study a family of stochastic differential equations with two parameters. Within this family, we have ODEs on a Lie group and Brownian motions ( more generally hypoelliptic diffusions) on a subgroup. For the reason we explain below these equations are called stochastic interpolation equations.Date: Monday, November 09, 2015
Let us now assume a reductive decomposition of the Lie algebras. We take one of the parameter to zero and observe a conserved quantity, taking values in a non-linear space. After reduction, we obtain a slow motion which converges, after a suitable scaling, to either a smooth curve or to a diffusion process on the orbit manifold. For Berger’s spheres with the torus subgroup we obtain a scaled Brownian motion, for SO(n) we see a scaled Brownian motion on the unit n-sphere. The scale for the Brownian motion depends on an eigenvalue of the hypoelliptic operator.