A stochastic heat equation with non-locally Lipschitz coefficients
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908473905643520 |
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| author | Chen, Le Huang, Jingyu Tao, Wenxuan |
| author_facet | Chen, Le Huang, Jingyu Tao, Wenxuan |
| contents | We consider the stochastic heat equation (SHE) on the torus $\mathbb{T}=[0,1]$, driven by space-time white noise $\dot W$, with an initial condition $u_0$ that is nonnegative and not identically zero:
\begin{equation*}
\frac{\partial u}{\partial t} = \tfrac{1}{2}\frac{\partial^2 u}{\partial x^2} + b(u) + σ(u)\dot{W}.
\end{equation*} The drift $b$ and diffusion coefficient $σ$ are Lipschitz continuous away from zero, although their Lipschitz constants may blow up as the argument approaches zero. We establish the existence of a unique global mild solution that remains strictly positive. Examples include $b(u)=u|\log u|^{A_1}$ and $σ(u)=u|\log u|^{A_2}$ with $A_1\in(0,1)$ and $A_2\in(0,1/4)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_23637 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A stochastic heat equation with non-locally Lipschitz coefficients Chen, Le Huang, Jingyu Tao, Wenxuan Probability 2020: Primary 60H15, Secondary 35R60 We consider the stochastic heat equation (SHE) on the torus $\mathbb{T}=[0,1]$, driven by space-time white noise $\dot W$, with an initial condition $u_0$ that is nonnegative and not identically zero: \begin{equation*} \frac{\partial u}{\partial t} = \tfrac{1}{2}\frac{\partial^2 u}{\partial x^2} + b(u) + σ(u)\dot{W}. \end{equation*} The drift $b$ and diffusion coefficient $σ$ are Lipschitz continuous away from zero, although their Lipschitz constants may blow up as the argument approaches zero. We establish the existence of a unique global mild solution that remains strictly positive. Examples include $b(u)=u|\log u|^{A_1}$ and $σ(u)=u|\log u|^{A_2}$ with $A_1\in(0,1)$ and $A_2\in(0,1/4)$. |
| title | A stochastic heat equation with non-locally Lipschitz coefficients |
| topic | Probability 2020: Primary 60H15, Secondary 35R60 |
| url | https://arxiv.org/abs/2507.23637 |