[Paper Review] On some properties of a class of fractional stochastic heat equations

Mohammud Foondun, Wei Liu+1 more|arXiv (Cornell University)|Apr 27, 2014

Stochastic processes and financial applicationsEconomics, Econometrics and Finance5 references3 citations

TL;DR

This paper investigates the growth of second moments in fractional stochastic heat equations driven by space-time white noise or spatially correlated noise, with the generator linked to an $α$-stable process killed upon exiting a ball. Using sharp heat kernel estimates and renewal-type inequalities involving the Mittag-Leffler function, it establishes that the excitation index—measuring the logarithmic growth rate of solution energy with respect to noise intensity—scales as $\frac{2\alpha}{\alpha-1}$ for white noise and $\frac{2\alpha}{\alpha-\beta}$ for colored noise with Riesz kernel correlation, extending prior results to a broader class of Lévy-driven SPDEs.

ABSTRACT

We consider nonlinear parabolic stochastic equations of the form $\partial_t u=\sL u + λσ(u)\dot ξ$ on the ball $B(0,\,R)$, where $\dot ξ$ denotes some Gaussian noise and $σ$ is Lipschitz continuous. Here $\sL$ corresponds to an $α$-stable process killed upon exiting $B(0, R)$. We will consider two types of noise; space-time white noise and spatially correlated noise. Under a linear growth condition on $σ$, we study growth properties of the second moment of the solutions.

Motivation & Objective

• To extend the understanding of noise excitability in stochastic heat equations beyond the classical case of Brownian motion and space-time white noise.
• To analyze the asymptotic behavior of the second moment of solutions to fractional SPDEs driven by general Lévy noise, particularly $α$-stable processes.
• To establish precise logarithmic growth rates of solution energy in terms of noise intensity $λ$, generalizing known results for the standard heat equation.
• To develop and apply new analytical techniques involving Dirichlet heat kernel estimates and renewal inequalities with Mittag-Leffler functions.

Proposed method

• The study employs mild solution formulations via integral equations driven by the fractional Dirichlet heat kernel $p_D(t,x,y)$ associated with $α$-stable processes killed at the boundary.
• Sharp heat kernel estimates for killed $α$-stable processes are used to compare with the un-killed case, enabling control over solution growth.
• Renewal-type inequalities are derived and analyzed using the asymptotic properties of the Mittag-Leffler function, crucial for bounding moments.
• A localized version of the method from [11] is adapted to handle the more complex correlation structure of spatially colored noise.
• The analysis is extended to various generators, including censored stable, relativistic stable, and sum of fractional Laplacians, by verifying matching short-time heat kernel behavior.
• The excitation index is defined and computed via the limit $\lim_{\lambda \to \infty} \frac{\log \log \mathcal{E}_t(\lambda)}{\log \lambda}$, where $\mathcal{E}_t(\lambda)$ is the $L^2$-norm of the solution.

Experimental results

Research questions

• RQ1 How does the second moment of the solution to a fractional stochastic heat equation grow as the noise intensity $\lambda$ tends to infinity?
• RQ2 What is the precise scaling of the excitation index when the noise is space-time white versus spatially correlated with a Riesz kernel of order $\beta$?
• RQ3 Can the results from the classical case ($\alpha=2$) be extended to general $\alpha$-stable processes with $\alpha \in (1,2)$?
• RQ4 How do different boundary conditions and generators (e.g., censored, relativistic, sum of fractional Laplacians) affect the growth rate of solution energy?
• RQ5 What analytical tools are necessary to handle the non-Markovian and long-range correlated nature of the noise in such SPDEs?

Key findings

• For space-time white noise, the excitation index of the solution is $\frac{2\alpha}{\alpha-1}$, recovering the known $\exp(\lambda^4)$ growth when $\alpha=2$.
• For spatially correlated noise with Riesz kernel $f(x,y) = |x-y|^{-\beta}$, the excitation index is $\frac{2\alpha}{\alpha - \beta}$, which depends on both the stability index $\alpha$ and correlation strength $\beta$.
• The growth rate of $\mathbb{E}|u_t(x)|^2$ as $\lambda \to \infty$ satisfies $\lim_{\lambda \to \infty} \frac{\log \log \mathbb{E}|u_t(x)|^2}{\log \lambda} = \frac{2\alpha}{\alpha - 1}$ for $x$ in the interior of the domain.
• The same growth rate $\frac{2\alpha}{\alpha - \beta}$ is obtained for censored stable processes, relativistic stable processes, and sums of fractional Laplacians, provided the short-time heat kernel behaves like $t^{-d/\alpha}$.
• The method relies on sharp heat kernel estimates and the asymptotic behavior of the Mittag-Leffler function to control moment growth through renewal inequalities.
• The results are robust across different generators, as long as the heat kernel satisfies $p(t,x,y) \asymp t^{-d/\alpha}$ for $|x-y| \leq t^{1/\alpha}$, which holds for a wide class of Lévy processes in bounded domains.