# 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 language English 133–158 26 Stochastic and Partial Differential Equations: Analysis and Computations 3 2 https://doi.org/10.1007/s40072-015-0045-y Published - 1 Jun 2015

### Keywords

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