Sharp Large Deviation Estimates for the Stochastic Heat Equation

Abstract

We consider the family {Xe, e≥0} of solution to the heat equation on [0,T]×[0,1] perturbed by a small space-time white noise, that is ∂tXe=\(\partial _{x,x}^2 \)Xe+b({Xe})+eσ({Xe})\(\dot W\). Then, for a large class of Borelian subsets of the continuous functions on [0,T]×[0,1], we get an asymptotic expansion of P({Xe}∈A) as e↘0. This kind of expansion has been handled for several stochastic systems, ranging from Wiener integrals to diffusion processes.