# 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

1 Citation (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.
Language English 133–158 26 Stochastic and Partial Differential Equations: Analysis and Computations 3 2 10.1007/s40072-015-0045-y Published - 1 Jun 2015

### Fingerprint

Stochastic Heat Equation
Fractional Brownian Motion
KPZ Equation
Space-time White Noise
Denote
Stochastic Partial Differential Equations
Laplace Operator
Fractional
Partial

### Keywords

• stochastic heat equation
• fractional Brownian motion
• KPZ equation

### Cite this

title = "Analysis of the gradient of the solution to a stochastic heat equation via fractional Brownian motion",
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.",
keywords = "stochastic heat equation, fractional Brownian motion, gradient estimates, KPZ equation",
author = "Mohammud Foondun and Davar Khoshnevisan and Pejman Mahboubi",
year = "2015",
month = "6",
day = "1",
doi = "10.1007/s40072-015-0045-y",
language = "English",
volume = "3",
pages = "133–158",
journal = "Stochastic and Partial Differential Equations: Analysis and Computations",
issn = "2194-0401",
publisher = "Springer",
number = "2",

}

Analysis of the gradient of the solution to a stochastic heat equation via fractional Brownian motion. / Foondun, Mohammud; Khoshnevisan, Davar; Mahboubi, Pejman.

In: Stochastic and Partial Differential Equations: Analysis and Computations, Vol. 3, No. 2, 01.06.2015, p. 133–158.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Analysis of the gradient of the solution to a stochastic heat equation via fractional Brownian motion

AU - Foondun, Mohammud

AU - Khoshnevisan, Davar

AU - Mahboubi, Pejman

PY - 2015/6/1

Y1 - 2015/6/1

N2 - 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.

AB - 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.

KW - stochastic heat equation

KW - fractional Brownian motion

KW - KPZ equation

U2 - 10.1007/s40072-015-0045-y

DO - 10.1007/s40072-015-0045-y

M3 - Article

VL - 3

SP - 133

EP - 158

JO - Stochastic and Partial Differential Equations: Analysis and Computations

T2 - Stochastic and Partial Differential Equations: Analysis and Computations

JF - Stochastic and Partial Differential Equations: Analysis and Computations

SN - 2194-0401

IS - 2

ER -