The objective of this talk is to outline one general method of proving the Large Deviations Principle (LDP) for stochastic partial differential equations (SPDEs).
SPDEs are a rich source of asymptotic problems, including large deviations, because of many dufferent random perturbations that can appear in such equations and numerous function spaces in which the solutions of the equations can be studied.
A standard way to establish LDP for a stochastic equation driven by a Brownian motion is to use the contraction, or continuous mapping, principle. The key step in the proof is to show that the solution of the equation is a continuous functional of the Brownian motion, which is often difficult in the case of SPDEs.
With the martingale approach, the LDP is derived from the local large deviations principle combined with the exponential tightness of the family of the solutions. The approach works in any Banach function space, and the proof of the local LDP often leads to an explicit expression for the action functional. The martingale approach has been proven an effective alternative to the contraction principle in the case of stochastic ordinary differential equations, but so far has never been applied to SPDEs.
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