### Abstract

Language | English |
---|---|

Pages | 133–158 |

Number of pages | 26 |

Journal | Stochastic and Partial Differential Equations: Analysis and Computations |

Volume | 3 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Jun 2015 |

### Fingerprint

### Keywords

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

### Cite this

*Stochastic and Partial Differential Equations: Analysis and Computations*,

*3*(2), 133–158. https://doi.org/10.1007/s40072-015-0045-y

}

*Stochastic and Partial Differential Equations: Analysis and Computations*, vol. 3, no. 2, pp. 133–158. https://doi.org/10.1007/s40072-015-0045-y

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

Research output: Contribution to journal › Article

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 - gradient estimates

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 -