Analysis of the gradient of the solution to a stochastic heat equation\n via fractional Brownian motion

2014/06/19 by Mohammud Foondun, Davar Khoshnevisan, Foondun, Mohammud +3 · 5 citations
Computer Science · Economics, Econometrics and Finance · Mathematics · #47B80 #60G17 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Fractional Differential Equations Solutions #Mathematical Biology Tumor Growth #Nonlinear Partial Differential Equations #Primary. 60H15 #Probability (math.PR) #Secondary. 60H10 #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1406.5246

openalex publication_date 2014/06/19 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28

Abstract

Consider the stochastic partial differential equation \∂_t u =\nLu+\σ(u)\ξ, where \ξ denotes space-time white noise and\nL:=-(-\Δ)^\α/2 denotes the fractional Laplace operator of index\n\α/2\∈( nicefrac12 ,,1]. We study the detailed behavior of the\napproximate spatial gradient u_t(x)-u_t(x-\ε) at fixed times t>0,\nas \ε downarrow0. We discuss a few applications of this work to the\nstudy of the sample functions of the solution to the KPZ equation as well.\n

Cited by

• Uniform dimension theorems for parabolic SPDEs
• Local linearization for estimating the diffusion parameter of nonlinear stochastic wave equations with spatially correlated noise
• Temporal properties of the stochastic fractional heat equation with rough dependence in space
• Sharp upper bounds on hitting probabilities for the solution to the stochastic heat equation on the line
• On the spatio-temporal increments of nonlinear parabolic SPDEs and the open KPZ equation

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