Bill Faris: The Ornstein-Uhlenbeck process in infinite dimensions

The Ornstein-Uhlenbeck process may be characterized as a stochastic process that is centered, Gaussian, Markov, stationary, and mean continuous. (Think of a model of the velocity of a tagged particle in a gas at equilibrium.) This talk describes classical work by Ito that characterizes all such processes, even in the infinite dimensional situation. It will present an elementary construction of the general process. The Euclidean free field leads to a particularly natural infinite dimensional example. 

 

There are two rather different variants of the Ornstein-Uhlenbeck process.  In one, there is a balance between random diffusion and linear drift that results in a stationary distribution. The other is a deterministic linear dynamical system with an invariant stationary distribution. Why treat these two variants together? Because, as will be shown, they are inescapably connected.