On the solutions of nonlinear stochastic fractional partial differential equations in one spatial dimension

Analysis of the gradient of the solution to a stochastic heat equation via fractional Brownian motion

“…We assume also that σ is Lipschitz continuous; that is, Lip σ < ∞, where E |u t (x)| k < ∞ for all T > 0, (1.2) for one, hence all, k ∈ [2 , ∞); see [7]. The main objective of this paper is to study the spatial gradient of the random function x → u t (x) where the time parameter t > 0 is held fixed.…”

“…This bound follows essentially from the Appendix of [9]; see also [7]. Therefore, we will not describe a proof.…”

Stochastic fractional partial differential equations driven by Poisson white noise

“…It is selfadjoint only when δ = 0 and in this case, it coincides with the fractional power of the Laplacian. When α = 2 it is the Laplacian itself (see [4,5]). …”

“…Let us recall some well-known properties, (see [5]), of the Green kernel G α which will be used later.…”

“…These operators generate symmetric stable semigroups when the order of derivation is less than 2. Our idea is inspired by the paper of Debbi and Dozzi [5] in which the authors studied a nonlinear stochastic fractional partial differential equations of high order containing also derivatives of entire order and perturbed by Gaussian white noise. The aim of this paper is to generalize the result of [1] to nonlinear stochastic fractional partial differential equations.…”