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- Title
Analysis of the gradient for the stochastic fractional heat equation with spatially-colored noise in $ \mathbb{R}^d $.
- Authors
Wang, Ran
- Abstract
Consider the stochastic partial differential equation$ \begin{equation*} \frac{\partial }{\partial t}u_t({\boldsymbol {x}}) = -(-\Delta)^{\frac{\alpha}{2}}u_t({\boldsymbol {x}}) +b\left(u_t({\boldsymbol {x}})\right)+\sigma\left(u_t({\boldsymbol {x}})\right) \dot F(t, {\boldsymbol {x}}), \ \ \ t\ge0, {\boldsymbol {x}}\in \mathbb{R}^d, \end{equation*} $where $ -(-\Delta)^{\frac{\alpha}{2}} $ denotes the fractional Laplacian with the power $ \alpha/2\in (1/2, 1] $, and the driving noise $ \dot F $ is a centered Gaussian random field which is white in time and with a spatial homogeneous covariance given by the Riesz kernel. We study the detailed behavior of the approximation spatial gradient $ u_t({\boldsymbol {x}})-u_t({\boldsymbol {x}}-{{\varepsilon}} \boldsymbol e) $ at any fixed time $ t>0 $, as $ {{\varepsilon}}\downarrow 0 $, where $ \boldsymbol e $ is a unit vector in $ \mathbb{R}^d $. As applications, we deduce the law of iterated logarithm and the behavior of the $ q $-variations of the solution in space.
- Subjects
STOCHASTIC analysis; FRACTIONAL powers; SPATIAL behavior; RANDOM fields; NOISE; HEAT equation
- Publication
Discrete & Continuous Dynamical Systems - Series B, 2024, Vol 29, Issue 6, p1
- ISSN
1531-3492
- Publication type
Article
- DOI
10.3934/dcdsb.2023201