Stochastic fractional heat equation with general rough noise
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917394466734080 |
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| author | Qian, Bin Wang, Ran |
| author_facet | Qian, Bin Wang, Ran |
| contents | Consider the following nonlinear one-dimensional stochastic fractional heat equation $$\frac{\partial }{\partial t}u(t, x)= -(-Δ)^{α/2}u(t, x) +σ(t,x,u(t,x)) \dot{W}(t, x), $$
where $-(-Δ)^{α/2}$ is the fractional Laplacian on $\mathbb R$ for $1<α<2$, and $\dot{W}$ is a Gaussian noise that is white in time and behaves in space as a fractional Brownian motion with Hurst index $H$ satisfying $\frac{3-α}{4}<H<\frac12$.
When $α=2$, Hu and Wang ({\it Ann. Inst. Henri Poincaré Probab. Stat.} {\bf 58} (2022) 379-423) studied the well-posedness of the solution and its Hölder continuity, removing the technical condition $σ(0)=0$ that was previously assumed in Hu et al. ({\it Ann. Probab.} {\bf 45} (2017) 4561-4616). Their approach relied on working in a weighted space with a suitable power decay function.
For the case $α\in (1,2)$, inspired by Hu and Wang, we investigate the well-posedness of the stochastic fractional heat equation without imposing the technical condition of $σ(0)=0$, which was required in the earlier work of Liu and Mao ({\it Bull. Sci. Math.} {\bf181} (2022) 103207). In our analysis, precise estimates of the heat kernel associated with the fractional Laplacian $-(-Δ)^{α/2}$ play a crucial role. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_07697 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Stochastic fractional heat equation with general rough noise Qian, Bin Wang, Ran Probability 60H15, 60G15, 60G22 Consider the following nonlinear one-dimensional stochastic fractional heat equation $$\frac{\partial }{\partial t}u(t, x)= -(-Δ)^{α/2}u(t, x) +σ(t,x,u(t,x)) \dot{W}(t, x), $$ where $-(-Δ)^{α/2}$ is the fractional Laplacian on $\mathbb R$ for $1<α<2$, and $\dot{W}$ is a Gaussian noise that is white in time and behaves in space as a fractional Brownian motion with Hurst index $H$ satisfying $\frac{3-α}{4}<H<\frac12$. When $α=2$, Hu and Wang ({\it Ann. Inst. Henri Poincaré Probab. Stat.} {\bf 58} (2022) 379-423) studied the well-posedness of the solution and its Hölder continuity, removing the technical condition $σ(0)=0$ that was previously assumed in Hu et al. ({\it Ann. Probab.} {\bf 45} (2017) 4561-4616). Their approach relied on working in a weighted space with a suitable power decay function. For the case $α\in (1,2)$, inspired by Hu and Wang, we investigate the well-posedness of the stochastic fractional heat equation without imposing the technical condition of $σ(0)=0$, which was required in the earlier work of Liu and Mao ({\it Bull. Sci. Math.} {\bf181} (2022) 103207). In our analysis, precise estimates of the heat kernel associated with the fractional Laplacian $-(-Δ)^{α/2}$ play a crucial role. |
| title | Stochastic fractional heat equation with general rough noise |
| topic | Probability 60H15, 60G15, 60G22 |
| url | https://arxiv.org/abs/2604.07697 |