### 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 |
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Pages (from-to) | 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 |

### Keywords

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

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## Cite this

Foondun, M., Khoshnevisan, D., & Mahboubi, P. (2015). Analysis of the gradient of the solution to a stochastic heat equation via fractional Brownian motion.

*Stochastic and Partial Differential Equations: Analysis and Computations*,*3*(2), 133–158. https://doi.org/10.1007/s40072-015-0045-y