Temporal regularity for the stochastic heat equation with rough dependence in space
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866918130744295424 |
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| author | Qian, Bin Wang, Min Wang, Ran Xiao, Yimin |
| author_facet | Qian, Bin Wang, Min Wang, Ran Xiao, Yimin |
| contents | Consider the nonlinear stochastic heat equation $$
\frac{\partial u (t,x)}{\partial t}=\frac{\partial^2 u (t,x)}{\partial x^2}+ σ(u (t,x))\dot{W}(t,x),\quad t> 0,\,
x\in \mathbb{R}, $$ where $\dot W$ is a Gaussian noise which is white in time and has the covariance of a fractional Brownian motion with Hurst parameter $H\in(\frac 14,\frac 12)$ in the space variable. When $σ(0)=0$, the well-posedness of the solution and its Hölder continuity have been proved by Hu et al. \cite{HHLNT2017}. In this paper, we study the asymptotic properties of the temporal gradient $u(t+\varepsilon, x)-u(t, x)$ at any fixed $t \ge 0$ and $x\in \mathbb R$, as $\varepsilon\downarrow 0$. As applications, we deduce Khintchine's law of iterated logarithm, Chung's law of iterated logarithm, and a result on the $q$-variations of the temporal process $\{u(t, x)\}_{t \ge 0}$, where $x\in \mathbb R$ is fixed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_03864 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Temporal regularity for the stochastic heat equation with rough dependence in space Qian, Bin Wang, Min Wang, Ran Xiao, Yimin Probability 60H15, 60G17, 60G22 Consider the nonlinear stochastic heat equation $$ \frac{\partial u (t,x)}{\partial t}=\frac{\partial^2 u (t,x)}{\partial x^2}+ σ(u (t,x))\dot{W}(t,x),\quad t> 0,\, x\in \mathbb{R}, $$ where $\dot W$ is a Gaussian noise which is white in time and has the covariance of a fractional Brownian motion with Hurst parameter $H\in(\frac 14,\frac 12)$ in the space variable. When $σ(0)=0$, the well-posedness of the solution and its Hölder continuity have been proved by Hu et al. \cite{HHLNT2017}. In this paper, we study the asymptotic properties of the temporal gradient $u(t+\varepsilon, x)-u(t, x)$ at any fixed $t \ge 0$ and $x\in \mathbb R$, as $\varepsilon\downarrow 0$. As applications, we deduce Khintchine's law of iterated logarithm, Chung's law of iterated logarithm, and a result on the $q$-variations of the temporal process $\{u(t, x)\}_{t \ge 0}$, where $x\in \mathbb R$ is fixed. |
| title | Temporal regularity for the stochastic heat equation with rough dependence in space |
| topic | Probability 60H15, 60G17, 60G22 |
| url | https://arxiv.org/abs/2501.03864 |