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arXiv · 1406.5246

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

Mohammud Foondun·Davar Khoshnevisan·Pejman Mahboubi
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Abstract

Consider the stochastic partial differential equation $\partial_t u = Lu+σ(u)ξ$, where $ξ$ denotes space-time white noise and $L:=-(-Δ)^{α/2}$ denotes the fractional Laplace operator of index $α/2\in(\nicefrac12\,,1]$. We study the detailed behavior of the approximate spatial gradient $u_t(x)-u_t(x-\varepsilon)$ at fixed times $t>0$, as $\varepsilon\downarrow0$. We discuss a few applications of this work to the study of the sample functions of the solution to the KPZ equation as well.

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Mohammud Foondun, Davar Khoshnevisan, Pejman Mahboubi. 2014-06-20. Analysis of the gradient of the solution to a stochastic heat equation via fractional Brownian motion. https://arxiv.org/abs/1406.5246

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