Vasileios Maroulas (UNC)

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.
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Bartłomiej Siudeja (Purdue)

As a starting point I will discuss an exit time from cones for Brownian motion and stable processes. Next, several generalizations will be given along with couple questions about how general results could we expect. Finally, I would like to present a new result obtained using precise bounds for the transition densities of killed process. This approach is different then a usual way involving harmonic analysis.
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Sourav Chatterjee (Berkeley)

The Komlos-Major-Tusnady embedding theorem, that gives the `best possible' coupling of a discrete random walk with a Brownian motion, is widely considered to be one of the landmark results in probability theory and also possibly one of its most important. The proof, however, is notoriously heavy-handed and hard to check. In this talk I will present a soft functional-analytic proof of this theorem for the case of the simple random walk. This new proof, unlike the old one, seems generalizable to situations involving complex dependencies as in models from statistical physics (I will give some examples from an ongoing work).
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