Abstract. In this paper, one investigates the following type of transportation-information T c I inequalities: α(T c (ν, µ)) ≤ I(ν|µ) for all probability measures ν on some metric space (X , d), where µ is a given probability measure, T c (ν, µ) is the transportation cost from ν to µ with respect to some cost function c(x, y) on 2 . It is proved that W 2 I is stronger than Poincaré inequality, weaker than log-Sobolev inequality, and equivalent to it when Bakry-Emery's curvature is bounded from below. For the trivial metric cost d, one establishes the sharp transportation-information inequality W 1
In this paper, we prove a central limit theorem and establish a moderate deviation principle for a perturbed stochastic wave equation defined on r0, T sˆR 3 . This equation is driven by a Gaussian noise, white in time and correlated in space. The weak convergence approach plays an important role.
Via Φ-Sobolev inequalities, we give some sharp integrability conditions on F for the large deviation principle of the empirical mean 1 T T 0 F (X s )ds for large time T , where F is unbounded with values in some separable Banach space. Several examples are provided.
In this paper, we study the Moderate Deviation Principle for a perturbed stochastic heat equation in the whole space R d , d ě 1. This equation is driven by a Gaussian noise, white in time and correlated in space, and the differential operator is a fractional derivative operator. The weak convergence method plays an important role.
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