On a fractional stochastic heat equation arising from the disordered pinning model

TL;DR

This paper investigates the existence, uniqueness, and properties of solutions to a fractional stochastic heat equation with Gaussian noise, revealing conditions under which solutions are local or global, and exploring their positivity in the context of disordered pinning models.

Contribution

It provides new results on the existence and uniqueness of solutions for a fractional stochastic heat equation with Gaussian noise, especially highlighting the critical role of the fractional order and Hurst parameter.

Findings

Local $L^2$-solution exists for $ ho=2$.

Solutions are not $L^p$-integrable for $p>1$ when $ ho eq 2$.

Unique global $L^1$-solution exists under certain conditions on $ ho$ and $H$.

Abstract

We study the mild Skorohod solution to the following fractional stochastic heat equation on $\mathbb{R}$: \begin{equation} \begin{cases} \partial_t u(t,x)=-(-\Delta)^{\rho/2} u(t,x) +\beta u(t,x)\delta_0(x)\xi(t),\\ u(0,\cdot)=u_0(x), \end{cases} \end{equation} where $-(-\Delta)^{\rho/2}$ with $\rho\in(0,2]$ is the fractional Laplacian and $\xi$ is a Gaussian noise with covariance $\mathbb{E}[\xi(t) \xi(s)]=|t-s|^{2H-2}$ for $H\in(\frac12, 1]$. This equation with $\rho\in(1,2]$ arises naturally in the study of the disordered pinning model. We show that the equation admits a local $L^2$-solution when $\rho = 2$, whereas, for $\rho \in (0,2)$, any solution--if it exists uniquely--cannot be $L^p$-integrable for any $p > 1$. Moreover, inspired by the recent work of Quastel, Ramirez and Vir\'{a}g, we prove that the equation has a unique global $L^1$-solution whenever $\frac{1}{\rho}+1<2H$. We also establish the strict positivity of the solution. Our work partially fills the gap in the study of the Weinrib-Halperin prediction.