On a fractional stochastic heat equation arising from the disordered pinning model
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915828242317312 |
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| author | Li, Zi'an Song, Jian Wei, Ran Zhang, Hang |
| author_facet | Li, Zi'an Song, Jian Wei, Ran Zhang, Hang |
| contents | 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)=-(-Δ)^{ρ/2} u(t,x) +βu(t,x)δ_0(x)ξ(t),\\ u(0,\cdot)=u_0(x), \end{cases} \end{equation} where $-(-Δ)^{ρ/2}$ with $ρ\in(0,2]$ is the fractional Laplacian and $ξ$ is a Gaussian noise with covariance $\mathbb{E}[ξ(t) ξ(s)]=|t-s|^{2H-2}$ for $H\in(\frac12, 1]$. This equation with $ρ\in(1,2]$ arises naturally in the study of the disordered pinning model.
We show that the equation admits a local $L^2$-solution when $ρ= 2$, whereas, for $ρ\in (0,2)$, 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 Virág, we prove that the equation has a unique global $L^1$-solution whenever $\frac{1}ρ+1<2H$. We also establish the strict positivity of the solution. Our work partially fills the gap in the study of the Weinrib-Halperin prediction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_01823 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On a fractional stochastic heat equation arising from the disordered pinning model Li, Zi'an Song, Jian Wei, Ran Zhang, Hang Probability 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)=-(-Δ)^{ρ/2} u(t,x) +βu(t,x)δ_0(x)ξ(t),\\ u(0,\cdot)=u_0(x), \end{cases} \end{equation} where $-(-Δ)^{ρ/2}$ with $ρ\in(0,2]$ is the fractional Laplacian and $ξ$ is a Gaussian noise with covariance $\mathbb{E}[ξ(t) ξ(s)]=|t-s|^{2H-2}$ for $H\in(\frac12, 1]$. This equation with $ρ\in(1,2]$ arises naturally in the study of the disordered pinning model. We show that the equation admits a local $L^2$-solution when $ρ= 2$, whereas, for $ρ\in (0,2)$, 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 Virág, we prove that the equation has a unique global $L^1$-solution whenever $\frac{1}ρ+1<2H$. We also establish the strict positivity of the solution. Our work partially fills the gap in the study of the Weinrib-Halperin prediction. |
| title | On a fractional stochastic heat equation arising from the disordered pinning model |
| topic | Probability |
| url | https://arxiv.org/abs/2603.01823 |