On The L{2}-Solutions of Stochastic Fractional Partial Differential Equations; Existence, Uniqueness and Equivalence of Solutions

The aim of this work is to prove existence and uniqueness of $L^{2}-$solutions of stochastic fractional partial differential equations in one spatial dimension. We prove also the equivalence between several notions of $L^{2}-$solutions. The Fourier transform is used to give meaning to SFPDEs. This method is valid also when the diffusion coefficient is random

Mild Solutions for a Class of Fractional SPDEs and Their Sample Paths

In this article we introduce and analyze a notion of mild solution for a class of non-autonomous parabolic stochastic partial differential equations defined on a bounded open subset $D\subset\mathbb{R}^{d}$ and driven by an infinite-dimensional fractional noise. The noise is derived from an $L^{2}(D)$-valued fractional Wiener process $W^{H}$ whose covariance operator satisfies appropriate restrictions; moreover, the Hurst parameter $H$ is subjected to constraints formulated in terms of $d$ and the H\"{o}lder exponent of the derivative $h^\prime$ of the noise nonlinearity in the equations. We prove the existence of such solution, establish its relation with the variational solution introduced in \cite{nuavu} and also prove the H\"{o}lder continuity of its sample paths when we consider it as an $L^{2}(D)$--valued stochastic processes. When $h$ is an affine function, we also prove uniqueness. The proofs are based on a relation between the notions of mild and variational solution established in Sanz-Sol\'e and Vuillermot 2003, and adapted to our problem, and on a fine analysis of the singularities of Green's function associated with the class of parabolic problems we investigate. An immediate consequence of our results is the indistinguishability of mild and variational solutions in the case of uniqueness.Comment: 37 page

Symmetrization for fractional elliptic and parabolic equations and an isoperimetric application

We develop further the theory of symmetrization of fractional Laplacian operators contained in recent works of two of the authors. The theory leads to optimal estimates in the form of concentration comparison inequalities for both elliptic and parabolic equations. In this paper we extend the theory for the so-called \emph{restricted} fractional Laplacian defined on a bounded domain $\Omega$ of $\mathbb R^N$ with zero Dirichlet conditions outside of $\Omega$. As an application, we derive an original proof of the corresponding fractional Faber-Krahn inequality. We also provide a more classical variational proof of the inequality.Comment: arXiv admin note: substantial text overlap with arXiv:1303.297