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

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.