Growth rates for the Hölder coefficients of the linear stochastic fractional heat equation with rough dependence in space
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909034426138624 |
|---|---|
| author | Liu, Chang Qian, Bin Wang, Ran |
| author_facet | Liu, Chang Qian, Bin Wang, Ran |
| contents | We study the linear stochastic fractional heat equation $$
\frac{\partial}{\partial t}u(t,x)=-(-Δ)^{\fracα2}u (t,x)+\dot{W}(t,x),\ \ t> 0,\ \ x\in\RR, $$ where $-(-Δ)^{\fracα{2}}$ denotes the fractional Laplacian with power $α\in (1, 2)$, and the driving noise $\dot W$ is a centered Gaussian field which is white in time and has the covariance of a fractional Brownian motion with Hurst parameter $H\in\left(\frac {2-α}2,\frac 12\right)$. We establish exact asymptotics for the solution as both time and space variables tend to infinity and derive sharp growth rates for the Hölder coefficients. The proofs are based on Talagrand's majorizing measure theorem and Sudakov's minoration theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_22379 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Growth rates for the Hölder coefficients of the linear stochastic fractional heat equation with rough dependence in space Liu, Chang Qian, Bin Wang, Ran Probability 60H15, 60G17, 60G22 We study the linear stochastic fractional heat equation $$ \frac{\partial}{\partial t}u(t,x)=-(-Δ)^{\fracα2}u (t,x)+\dot{W}(t,x),\ \ t> 0,\ \ x\in\RR, $$ where $-(-Δ)^{\fracα{2}}$ denotes the fractional Laplacian with power $α\in (1, 2)$, and the driving noise $\dot W$ is a centered Gaussian field which is white in time and has the covariance of a fractional Brownian motion with Hurst parameter $H\in\left(\frac {2-α}2,\frac 12\right)$. We establish exact asymptotics for the solution as both time and space variables tend to infinity and derive sharp growth rates for the Hölder coefficients. The proofs are based on Talagrand's majorizing measure theorem and Sudakov's minoration theorem. |
| title | Growth rates for the Hölder coefficients of the linear stochastic fractional heat equation with rough dependence in space |
| topic | Probability 60H15, 60G17, 60G22 |
| url | https://arxiv.org/abs/2507.22379 |