| Given a stochastically forced dissipative PDE such as the 2D Navier
Stokes equations, the Ginzburg-Landau equations, or a reaction diffusion
equation; is the system Ergodic ? This question is fundamental to
justifying many assumptions made in modeling.
If it is ergodic at what rate does the system equilibrate. I will explain
how to address these questions. The analysis will suggest strategies to
analysis other properties of these SPDEs. The use of SPDEs in modeling
is common in fields ranging from turbulence to interface growth to population
dynamics to filtering.
In particular I will show that the stochastically forced 2D Navier Stokes
equations converges exponentially to a unique invariant measure.
I will discuss under what minimal conditions one should expect ergodic
behavior. The central ideas with be illustrated with simple model
systems.
Along the way I will discuss some issues which might be of interests
to a number of groups.
Probabilist: Coupling in an infinite dimensional Markov chain and in
a non-Markovian settings.
Exponential mixing, Harris chains, and Lyapunov functions.
Physicists: The description of the system through Gibbs measures and
the connections to classical facts from the theory of one dimensional phase
transitions. Making use of the enslaving of the high frequencies by the
low frequencies.
Analysts: Hypo-ellipticity of degenerate diffusions and why these systems
are "morally elliptic" even though at first glance they seem hypoelliptic.
Spectral gaps for diffusions on function spaces. |