"Ergodicity in infinite dimensions:
Degenerately forced Stochastic Partial Differential equations"
Jonathan Mattingly, Duke University
I will discuss a body of work which has emerged over the last number
of year to treat the ergodic theory of one class of Markov processes
on infinite dimensional phase spaces. In particular, I will discuss the
idea of an Asymptotically Strong Feller Diffusion. Using some
estimates from Malliavin calculus, we can apply the results to a class
of dissipative partial differential equations forced by only a few
brownian motions. This class includes the 2D Navier-Stokes equation
and reaction diffusion equations. This is an inherently nonlinear
phenomenon which uses the nonlinearity to move the randomness about
phase space. I will also describe how the theory give spectral gaps
in a Wasserstein metric for the Markov semigroups. The discussion will
center on phenomenon which are infinite dimensional in nature.