[Paper Review] Nonlinear stochastic time-fractional slow and fast diffusion equations on $\mathbb{R}^d$

Le Chen, Yaozhong Hu+1 more|arXiv (Cornell University)|Sep 25, 2015

Stochastic processes and financial applications3 citations

TL;DR

This paper establishes the existence, uniqueness, and moment bounds for solutions to nonlinear stochastic time-fractional diffusion equations on $\mathbb{R}^d$, driven by space-time white noise and involving Caputo time-fractional derivatives and Riemann-Liouville fractional integrals. The key contribution is proving these results under a generalized Dalang-type condition involving the parameters $\alpha$, $\beta$, $\gamma$, and dimension $d$, with solutions expressed via Fox H-functions and sample path regularity established for $\beta \in (0,1]$. The framework allows for measure-valued initial data and covers both slow ($\beta \in (0,1]$) and fast ($\beta \in (1,2)$) diffusion regimes.

ABSTRACT

This paper studies the nonlinear stochastic partial differential equation of fractional orders both in space and time variables: \[ \left(\partial^β+\fracν{2}(-Δ)^{α/2} ight)u(t,x) = I_t^γ\left[ρ(u(t,x))\dot{W}(t,x) ight],\quad t>0,\: x\in\mathbb{R}^d, \] where $\dot{W}$ is the space-time white noise, $α\in(0,2]$, $β\in(0,2)$, $γ\ge 0$ and $ν>0$. Fundamental solutions and their properties, in particular the nonnegativity, are derived. The existence and uniqueness of solution together with the moment bounds of the solution are obtained under Dalang's condition: $d<2α+\fracαβ\min(2γ-1,0)$. In some cases, the initial data can be measures. When $β\in (0,1]$, we prove the sample path regularity of the solution.

Motivation & Objective

• To extend the theory of stochastic partial differential equations to include time-fractional derivatives with general order $\beta \in (0,2)$ and fractional Laplacian diffusion of order $\alpha \in (0,2]$.
• To establish existence and uniqueness of solutions under a generalized Dalang-type condition involving $\alpha$, $\beta$, $\gamma$, and spatial dimension $d$, allowing for measure-valued initial data.
• To analyze the regularity of sample paths for the solution when $\beta \in (0,1]$, extending known results for the stochastic heat and wave equations.
• To derive fundamental solutions using the Fox H-function and characterize their properties, including nonnegativity and asymptotic behavior near the origin.
• To unify and generalize previous results on stochastic fractional diffusion, including the stochastic heat and wave equations, within a single fractional-order framework.

Proposed method

• The study employs the Caputo fractional derivative $\partial^\beta$ and Riemann-Liouville fractional integral $I_t^\gamma$ to model memory effects and noise smoothing in the stochastic PDE.
• The solution is represented as a stochastic integral equation involving convolution with fundamental solutions $Z(t,x)$, $Z^*(t,x)$, and $Y(t,x)$, which are expressed in terms of the Fox H-function.
• The existence and uniqueness of the solution are proven under the condition $d < 2\alpha + \frac{\alpha}{\beta}\min(2\gamma - 1, 0)$, a generalization of Dalang’s condition.
• The asymptotic behavior of the fundamental solutions near the origin is analyzed via residue calculus of the Fox H-function, determining the local singularities based on the parameters $\alpha$, $\beta$, $\gamma$, and $d$.
• Sample path regularity for $\beta \in (0,1]$ is established through moment bounds and Hölder continuity estimates derived from the structure of the Green's function.
• The framework allows for initial data to be measures, extending classical results that require $L^1$ or $L^2$ initial conditions.

Experimental results

Research questions

• RQ1 Under what conditions does a mild solution exist for the nonlinear stochastic time-fractional diffusion equation with space-time white noise on $\mathbb{R}^d$?
• RQ2 How does the interplay between the time-fractional order $\beta$, the spatial fractional order $\alpha$, and the noise smoothing parameter $\gamma$ affect the existence and moment bounds of the solution?
• RQ3 What is the precise asymptotic behavior of the fundamental solution near the origin, and how does it depend on the parameters $\alpha$, $\beta$, $\gamma$, and $d$?
• RQ4 Can the solution exhibit Hölder continuity in space and time when $\beta \in (0,1]$, and what is the sharp regularity threshold?
• RQ5 How do the results generalize known cases such as the stochastic heat equation ($\beta=1$, $\alpha=2$) and the stochastic wave equation ($\beta=2$, $\alpha=2$)?

Key findings

• The solution exists and is unique under the condition $d < 2\alpha + \frac{\alpha}{\beta}\min(2\gamma - 1, 0)$, which generalizes Dalang’s condition for SPDEs with time-fractional derivatives.
• Moment bounds for the solution are established, ensuring stochastic stability and providing a foundation for further analysis of intermittency and sample path behavior.
• For $\beta \in (0,1]$, the solution exhibits Hölder continuity in both space and time, with the regularity depending on the parameters $\alpha$, $\beta$, and $\gamma$.
• The fundamental solution is expressed using the Fox H-function, and its behavior near the origin is fully characterized through residue analysis, revealing power-law singularities of the form $x^{d/\alpha}$, $x^2$, or $x \log x$ depending on the parameter regime.
• The framework allows for initial data to be measures, extending the applicability of the theory beyond $L^p$-integrable initial conditions.
• In the limit $\gamma=0$, the existence condition becomes $d < 2\alpha + \frac{\alpha}{\beta}\min(-1, 0)$, and the solution exists only for $\beta > 2/3$ in the case $\alpha=2$, $d=1$, demonstrating the necessity of the fractional integral for regularity.