Analysis of the gradient for the stochastic fractional heat equation with spatially-colored noise in \mathbb R^d

2022/10/21 by Ran Wang, Wang, Ran
Computer Science · Economics, Econometrics and Finance · Mathematics · #60G17 #60H15 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Nonlinear Partial Differential Equations #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2210.11772

openalex publication_date 2022/10/21 · openalex created_date 2022/10/30 · openalex updated_date 2026/07/28

Abstract

Consider the stochastic partial differential equation (∂ )/(∂ t)u_t(x)= -(-Δ)^\fracα2u_t(x) +b(u_t(x))+σ(u_t(x)) F(t, x), t≥0, x∈ \mathbb R^d, where -(-Δ)^\fracα2 denotes the fractional Laplacian with the power α/2∈ (1/2,1], and the driving noise F is a centered Gaussian 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(x)-u_t(x-ε \mathbf e) at any fixed time t>0, as ε\downarrow 0, where \mathbf e is the 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.

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