Non-selfadjoint Ornstein-Uhlenbeck semigroups

The talk will describe Ornstein-Uhlenbeck semigroups and explore an example in infinite dimensions related to the renormalization group of quantum field theory. An Ornstein-Uhlenbeck semigroup describes a diffusion process with constant diffusion and linear drift. The resulting effect is Gaussian convolution followed by rescaling. The first goal of the talk is to contrast the situation of detailed balance, where the drift coefficient is self-adjoint, with a very different situation where the drift coefficient is skew-adjoint. It is shown that in the latter situation there can be a stationary measure, but only in infinite dimensions and only when the drift coefficient is given by an operator with absolutely continuous spectrum. The second goal of the talk is to show that one version of the renormalization group is given by a particular Ornstein-Uhlenbeck semigroup of this second type and to characterize this semigroup abstractly. The first part briefly reviews the theory of vector fields, that is, of autonomous systems of ordinary differential equations. Near an isolated zero of the vector field, that is, near a stationary solution of the system of differential equations, the equation is typically (but not always) equivalent to a system given by a linear vector field (a matrix equation). The second part is a similar review of the theory of stochastic differential equations driven by white noise (the derivative of a Wiener process). For such equations the solutions are random, but there are semigroups of linear operators that describe the evolution of expectations of functions of the process. In some circumstances there are stationary probability densities. One special case is when the equation satisfies the detailed balance condition. Then it is comparatively easy to compute the stationary probability densities. However, the processes described in this talk need not satisfy detailed balance. The third part introduces the main object of study, the Ornstein-Uhlenbeck processes and their associated semigroups. Such a process is defined by a stochastic differential equation given by a linear vector field and a non-degenerate