SEMINAIRE D'ANALYSE NUMERIQUE
Année universitaire 2010-2011
Jeudi 17 février 2011 :
Hitoshi ISHII
(Waseda University, Tokyo)
A PDE approach to small stochastic pertubations to Hamiltonian flows.
We present a PDE approach to the study of averaging principles for
(small) stochastic perturbations of Hamiltonian flows in 2D, which
is based on a recent joint work with P. E. Souganidis. Such problems were
introduced by Freidlin and Wentzel and have been the subject of extensive
study in the last several years. When the Hamiltonian flows has critical points
the averaging principle exhibits complicated behavior. Asymptotically
the slow (averaged) motion has 1D character and takes place on a graph,
and the question is to identify the limit motion in terms of PDE problems.
In their original work Freidlin and Wentzell, using probabilistic techniques,
considered perturbations by Brownian motions, while later Freidlin and Weber studied,
combining probabilistic and analytic techniques based on hypoelliptic operators, a special degenerate case. Recently Sowers revisited the uniformly elliptic case and
constructed what amounts to approximate correctors for the averaging
problem. Our approach is based on PDE techniques and applies
to general degenerate elliptic operators.
