Temporal properties of the stochastic fractional heat equation with spatially-colored noise

Authors:
Ran Wang and Yimin Xiao

Journal:
Theor. Probability and Math. Statist. **110** (2024), 121-142

MSC (2020):
Primary 60H15, 60G17, 60G22

DOI:
https://doi.org/10.1090/tpms/1209

Published electronically:
May 10, 2024

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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}), \quad t\ge 0,\: \boldsymbol {x}\in \mathbb R^d, \end{equation*} where $-(-\Delta )^{\frac {\alpha }{2}}$ denotes the fractional Laplacian with power $\frac {\alpha }{2}\in (\frac 12,1]$, and the driving noise $\dot F$ is a centered Gaussian field which is white in time and has a spatial homogeneous covariance given by the Riesz kernel. We study the detailed behavior of the approximation of the temporal gradient $u_{t+{\varepsilon }}(\boldsymbol {x})-u_t(\boldsymbol {x})$ at any fixed $t > 0$ and $\boldsymbol {x}\in \mathbb R^d$, as ${\varepsilon }\downarrow 0$. As applications, we deduce Khintchin’s law of iterated logarithm, Chung’s law of iterated logarithm, and a result on the $q$-variations of the temporal process $\{u_t(\boldsymbol {x})\}_{ t \ge 0}$ of the solution, where $\boldsymbol {x}\in \mathbb R^d$ is fixed.

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*Moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions*, Ann. Probab. **43** (2015), no. 6, 3006–3051. MR **3433576**
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*Analysis of stochastic partial differential equations*, American Mathematical Soc., 2014. MR **3222416**
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*Growth rate of certain Gaussian processes*, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Volume 2: Probability Theory, University of California Press, 1972. MR **402897**
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*An asymptotic $0$-$1$ behavior of Gaussian processes*, Ann. Math. Stat. **42** (1971), no. 6, 2029–2035. MR **307317**
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*Transportation inequalities under uniform metric for a stochastic heat equation driven by time-white and space-colored noise*, Acta Appl. Math. **170** (2020), 81–97. MR **4163229**
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*Singular integrals and differentiability properties of functions*, Princeton University Press, 1970. MR **290095**
- M. Talagrand,
*Lower classes of fractional Brownian motion*, J. Theor. Probab. **9** (1996), 191–213. MR **1371076**
- C. A. Tudor and Y. Xiao,
*Sample path properties of the solution to the fractional-colored stochastic heat equation*, Stoch. Dyn. **17** (2017), no. 1, 1750004. MR **3583991**
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*An introduction to stochastic partial differential equations*, Ècole d’Été de Probabilités de Saint-Flour, XIV–1984, Lecture Notes in Math. 1180, Springer, Berlin, 265–439, 1986. MR **876085**
- R. Wang,
*Analysis of the gradient for the stochastic fractional heat equation with spatially-colored noise in $\mathbb R^d$*, Discrete Contin. Dyn. Syst. B **29** (2024), no. 6, 2769–2785. MR **4729304**
- R. Wang and S. Zhang,
*Decompositions of stochastic convolution driven by a white-fractional Gaussian noise*, Front. Math. China **16** (2021), no. 4, 1063–1073. MR **4307364**

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Additional Information

**Ran Wang**

Affiliation:
School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, People’s Republic of China

Email:
rwang@whu.edu.cn

**Yimin Xiao**

Affiliation:
Department of Statistics and Probability, Michigan State University, East Lansing, Michigan 48824

MR Author ID:
256757

ORCID:
0000-0002-9474-1605

Email:
xiaoy@msu.edu

Keywords:
Stochastic heat equation,
fractional Brownian motion,
fractional Laplacian,
law of iterated logarithm,
$q$-variation

Received by editor(s):
July 26, 2023

Accepted for publication:
November 29, 2023

Published electronically:
May 10, 2024

Additional Notes:
The research of the second author was partially supported by NSF grant DMS-2153846.

Article copyright:
© Copyright 2024
Taras Shevchenko National University of Kyiv