No abstract

In this paper we prove a large deviation principle (ldp) for a perturbed stochastic heat equation defined on [0, T ] × [0, 1] d . This equation is driven by a Gaussian noise, white in time and correlated in space. Firstly, we show the Hölder continuity for the solution of the stochastic heat equation. Secondly, we check that our Gaussian process satisfies a ldp and some requirements on the skeleton of the solution. Finally, we prove the called Freidlin-Wentzell inequality. In order to obtain all these results we need precise estimates of the fundamental solution of this equation.

Abstract. We obtain a closed expression for the solution of a linear Volterra integral equation with an additive Hölder continuous noise, which is a fractional Young integral, and with a function as initial condition. This solution is given in terms of the Mittag-Leffler function. Then we study the stability of the solution via the fractional calculus. As an application we analyze the stability in the mean of some stochastic fractional integral equations with a functional of the fractional Brownian motion as an additive noise.

In this paper we deal with generalized fractional kinetic equations driven by a Gaussian noise, white in time and correlated in space, and where the diffusion operator is the composition of the Bessel and Riesz potentials for any fractional parameters. We give results on the existence and uniqueness of solutions by means of a weak formulation and study the Hölder continuity. Moreover, we prove the existence of a smooth density associated to the solution process and study the asymptotics of this density. Finally, when the diffusion coefficient is constant, we look for its Gaussian index.

This paper is devoted to a detailed study of a p-spins interaction model with external field, including some sharp bounds on the speed of self averaging of the overlap as well as a central limit theorem for its fluctuations, the thermodynamical limit for the free energy and the definition of an Almeida-Thouless type line. Those results show that the external field dominates the tendency to disorder induced by the increasing level of interaction between spins, and our system will share many of its features with the SK model, which is certainly not the case when the external magnetic field vanishes.Observe that we should write I = I(J), I c = I c (J), but we will omit this dependence for sake of readability. Using these sets, we defineProof: This proof is an extension of [9, Proposition 2.4.2], whose details are given for sake of completeness. We havewith κ n = n(n + 1). Note that, for any l ∈ {1, . . . , n},Thuswhere we recall that κ n = n(n + 1). This can be read asThe same kind of computations can be lead for A 1 , integrating first by parts with respect to the variables g I . In this case, we obtainSubstracting A 1 and A 2 , we get the desired result. As a consequence of the last proposition, we can bound ν k,t (f ) by ν k,1 (f ) as follows: 1 2 J∈Q p N,k z J ε l J , n l=1 ν k,t f ε l J ε n+1 J − n+1 l=1 ν k,t f ε l J ε n+1 J + n(n + 1)ν k,t f ε n+1 J ε n+2 J Now, from Lemma 8.4, we obtain easily u 2 N |Q p N,k | ≤ K N , that gives us the desired result.