Stochastic functional differential equations driven by L\'{e}vy processes and quasi-linear partial integro-differential equations

In this article we study a class of stochastic functional differential equations driven by L\'{e}vy processes (in particular, $\alpha$-stable processes), and obtain the existence and uniqueness of Markov solutions in small time intervals. This corresponds to the local solvability to a class of quasi-linear partial integro-differential equations. Moreover, in the constant diffusion coefficient case, without any assumptions on the L\'{e}vy generator, we also show the existence of a unique maximal weak solution for a class of semi-linear partial integro-differential equation systems under bounded Lipschitz assumptions on the coefficients. Meanwhile, in the nondegenerate case (corresponding to $\Delta^{\alpha/2}$ with $\alpha\in(1,2]$), based upon some gradient estimates, the existence of global solutions is established too. In particular, this provides a probabilistic treatment for the nonlinear partial integro-differential equations, such as the multi-dimensional fractal Burgers equations and the fractal scalar conservation law equations.Comment: Published in at http://dx.doi.org/10.1214/12-AAP851 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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

Weak convergence of Galerkin approximations for fractional elliptic stochastic PDEs with spatial white noise

The numerical approximation of the solution to a stochastic partial differential equation with additive spatial white noise on a bounded domain is considered. The differential operator is assumed to be a fractional power of an integer order elliptic differential operator. The solution is approximated by means of a finite element discretization in space and a quadrature approximation of an integral representation of the fractional inverse from the Dunford-Taylor calculus. For the resulting approximation, a concise analysis of the weak error is performed. Specifically, for the class of twice continuously Fr\'echet differentiable functionals with second derivatives of polynomial growth, an explicit rate of weak convergence is derived, and it is shown that the component of the convergence rate stemming from the stochasticity is doubled compared to the corresponding strong rate. Numerical experiments for different functionals validate the theoretical results.Comment: 22 pages, 1 figur

Large deviation principle for fractional Brownian motion with respect to capacity

We show that fractional Brownian motion(fBM) defined via Volterra integral representation with Hurst parameter $H\geq\frac{1}{2}$ is a quasi-surely defined Wiener functional on classical Wiener space,and we establish the large deviation principle(LDP) for such fBM with respect to $(p,r)$-capacity on classical Wiener space in Malliavin's sense