Small noise large deviations for infinite dimensional stochastic dynamical systems Public Deposited

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  • March 22, 2019
  • Maroulas, Vasileios
    • Affiliation: College of Arts and Sciences, Department of Statistics and Operations Research
  • Large deviations theory concerns with the study of precise asymptotics governing the decay rate of probabilities of rare events. A classical area of large deviations is the Freidlin–Wentzell (FW) theory that deals with path probability asymptotics for small noise Stochastic Dynamical Systems (SDS). For finite dimensional SDS, FW theory has been very well studied. The goal of the present work is to develop a systematic framework for the study of FW asymptotics for infinite dimensional SDS. Our first result is a general LDP for a broad family of functionals of an infinite dimensional small noise Brownian motion (BM). Depending on the application, the driving infinite dimensional BM may be given as a space–time white noise, a Hilbert space valued BM or a cylindrical BM. We provide sufficient conditions for LDP to hold for all such different model settings. As a first application of these results we study FW LDP for a class of stochastic reaction diffusion equations. The model that we consider has been widely studied by several authors. Two main assumptions imposed in all previous studies are the boundedness of the diffusion coefficient and a certain geometric condition on the underlying domain. These restrictive conditions are needed in proofs of certain exponential probability estimates that form the basis of classical proofs of LDPs. Our proofs instead rely on some basic qualitative properties, eg. existence, uniqueness, tightness, of certain controlled analogues of the original systems. As a result, we are able to relax the two restrictive requirements described above. As a second application, we study large deviation properties of certain stochastic diffeomorphic flows driven by an infinite sequence of i.i.d. standard real BMs. LDP for small noise finite dimensional flows has been studied by several authors. Typical space– time stochastic models with a realistic correlation structure in the spatial parameter naturally leads to infinite dimensional flows. We establish a LDP for such flows in the small noise limit. We also apply our result to a Bayesian formulation of an image analysis problem. An approximate maximum likelihood property is shown for the solution of an optimal image matching problem that involves the large deviation rate function.
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  • Budhiraja, Amarjit
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  • University of North Carolina at Chapel Hill
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