Space-time fractional diffusions in Gaussian noisy environment

Abstract

This paper studies the linear stochastic partial differential equation of fractional orders both in time and space variables $\left({\partial }^{\beta }+\frac{\nu }{2}(-\Delta {)}^{\alpha /2}\right)u(t,x)=\lambda u(t,x)\dot{W}(t,x)$, where $\dot{W}$ is a general Gaussian noise and $\beta \in (1/2,2)$, $\alpha \in (0,2]$. The existence and uniqueness of the solution, the moment bounds of the solution are obtained by using the fundamental solutions of the corresponding deterministic counterpart represented by the Fox H-functions. Along the way, we obtain some new properties of the fundamental solutions.

References in corpus (6)

• The fundamental solution of the space-time fractional diffusion equation
• Nonlinear stochastic time-fractional diffusion equations on $\mathbb{R}$: moments, Hölder regularity and intermittency
• Stochastic heat equation with rough dependence in space
• Asymptotic properties of some space-time fractional stochastic equations
• Intermittence and time fractional stochastic partial differential equations
• On a class of stochastic partial differential equations

Cited by in corpus (1)

• Intermittency fronts for space-time fractional stochastic partial differential equations in $(d+1)$ dimensions