Stochastic Interpolation Equation(概率论系列报告)
主 题: Stochastic Interpolation Equation(概率论系列报告)
报告人: Prof. Xue-mei Li (李雪梅) (The University of Warwick, UK)
时 间: 2015-08-10 15:30 - 16:30
地 点: 北京大学镜春园78号院77201
We present and study a stochastic interpolation equation inspired by the study of collapsing of a family of Riemannian manifolds to a lower dimensional manifolds. This model interpolates Brownian motions on a subgroup H of a Lie group G, and inhomogeneous scaling of Riemannian metrics. In the limit we obtain a pair of effective motions. This is reminscent of perturbed Hamiltonian systems, where the effective motion describes the trajectories of orbits over a very long time. Depending on the position of the perturbation vector, we have an averaged dynamical system or have `diffusion creation'. The study consists of three parts: (1) Reduction of complexity. The interpolation equation is a singularly perturbed stochastic equation. We explore the conservation law associated to the Brownian motion on H. (2) Determine the scale for the slow motion. Convergence of the slow variables. We use techniques from stochastic homogenization and randomly perturbed ODE’s. (3) Classify the effective limits. This talk focuses on part (3) and in particular on the question whether the effective motion is a Markov process. We use Dynkin’s criterion and explore the intrinsic symmetries. This last part is reminiscent of the following: the effective motion for randomly perturbed Hamiltonian systems is a diffusion on graphs.
