Vasileios Maroulas, University of North Carolina, Chapel Hill
TITLE:
Small noise large deviations for infinite dimensional stochastic dynamical systems
Abstract
Freidlin-Wentzell theory, one of the classical areas in large deviations,
deals with path probability asymptotics for small noise stochastic
dynamical systems. For finite dimensional stochastic differential
equations (SDE) there has been an extensive study of this problem.
In this work we are interested in infinite dimensional models, i.e.
the setting where the driving Brownian motion is infinite dimensional.
In recent years there has been lot of work on the study of large deviations
principle (LDP) for small noise infinite dimensional SDEs, much of which is
based on the ideas of Azencott (1980).
A key in this approach is obtaining suitable exponential tightness and
continuity estimates for certain approximations of the stochastic processes.
This becomes particularly hard in infinite dimensional setting where such
estimates are needed with metrics on exotic function spaces
(e.g. Hölder spaces, spaces of diffeomorphisms etc).
Our approach to the large deviation analysis is quite different and is based
on certain variational representation for infinite dimensional Brownian
motions. It bypasses all discretizations and finite dimensional
approximations and thus no exponential probability estimates are needed.
Proofs of LDP are reduced to demonstrating basic qualitative properties
(existence, uniqueness and tightness) of certain perturbations of the
original process. The approach has now been adopted by several authors
in recent works to study various infinite dimensional models such as
stochastic Navier-Stokes equations, stochastic flows of diffeomorphisms,
SPDEs with random boundary conditions.
As a first example of this approach, we consider a class of stochastic
reaction-diffusion equations, which have been studied by various authors.
We establish a large deviation principle under conditions that are
substantially weaker than those available in the literature.
We next study a family of stochastic flows of diffeomorphisms that arise
in certain image analysis problems. Large deviations for the case where
the driving noise is finite dimensional has been studied by Ben Arous
and Castell (1995). We extend these results to an infinite dimensional
setting and apply them to a problem of image analysis.
R E F R E S H M E N T S
4:00 to 4:25 PM
Kenneth W. Anderson
Memorial Reading Room
