Large deviations for stochastic heat equation with rough dependence in space

Abstract

Abstract. In this paper we establish a large deviation principle for the nonlinear onedimensional stochastic heat equation driven by a Gaussian noise which is white in time and which has the covariance of a fractional Brownian motion with Hurst parameter H ∈ ( 1 4 ,in the space variable.

A Large Deviation Principle for the Stochastic Heat Equation with General Rough Noise

“…However, the frameworks of [26] and [37] cannot be applied to SHE (1.1) because of the spatial rough noise with H P p 1 4 , 1 2 q. Comparing with the large deviation principle for SHE (1.1) under the assumption of σp0q " 0 in Hu et al [20], our result and its proof are in the weighted space Z p λ,T . Notice that the introduction of the weight brings many difficulties.…”

“…For the general nonlinear coefficient σ, which has a Lipschitz derivative and satisfies σp0q " 0, the existence and uniqueness of the solution were proved by Hu et al [17]. Under this condition, the large deviations, the moderate deviations and the transportation inequalities were studied in [20], [22], [9], respectively.…”

The law of the iterated logarithm for a class of SPDEs

“…Here we compare the two methods often applied in the literature to prove the LDP: the weak convergence approach, which was applied in [24] to prove the MDP for (1.1) and the Azencott method employed here to obtain the same result. For more examples of results on the LDP or MDP for SPDEs we refer the reader to [6,16,19,26,32,33,42,45,48] for the weak convergence approach and to [37,40,49] for the Azencott method. See Section 3 for more references for the Azencott method.…”

Large deviation principle for stochastic heat equation with general rough noise

“…For the general nonlinear coefficient σ, which has a Lipschitz derivative and satisfies σp0q " 0, the existence and uniqueness of the solution were proved by Hu et al [17]. Under this condition, the large deviations, the moderate deviations and the transportation inequalities were studied in [20], [22], [9], respectively.…”