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

Mohammud Foondun, Davar Khoshnevisan, Pejman Mahboubi

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

Consider the stochastic partial differential equation \(\partial _{t} u = Lu+\sigma (u)\xi \), where \(\xi \) denotes space–time white noise and \(L:=-(-\Delta )^{\alpha /2}\) denotes the fractional Laplace operator of index \(\alpha /2\in (1/2\,,1]\). We study the detailed behavior of the approximate spatial gradient \(u_{t}(x)-u_{t}(x-\varepsilon )\) at fixed times \(t>0\), as \(\varepsilon \downarrow 0\). We discuss a few applications of this work to the study of the sample functions of the solution to the KPZ equation as well.
Original languageEnglish
Pages (from-to)133–158
Number of pages26
JournalStochastic and Partial Differential Equations: Analysis and Computations
Volume3
Issue number2
DOIs
Publication statusPublished - 1 Jun 2015

Keywords

  • stochastic heat equation
  • fractional Brownian motion
  • gradient estimates
  • KPZ equation

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