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William G. Faris

Ornstein-Uhlenbeck and renormalization semigroups

Department of Mathematics
University of Arizona
Tucson, Arizona 85721 USA
email address: faris AT math.arizona.edu

The Ornstein-Uhlenbeck semigroup combines Gaussian diffusion
with the flow of a linear vector field. In infinite dimensional
settings there can be non-Gaussian invariant measures. This gives
a context for one version of the renormalization group. The
adjoint of the Ornstein-Uhlenbeck semigroup with respect to an
invariant measure need not be an Ornstein-Uhlenbeck
semigroup. This adjoint is the appropriate semigroup to analyze
the local stability of the invariant measure under the
renormalization group.