### Abstract

Ergodicity for local and nonlocal stochastic singular
$p$-Laplace equations is proven, without restriction on the spatial
dimension and for all $p\in[1,2)$. This generalizes previous results
from [B. Gess and J. M. Tölle,

*J. Math. Pures Appl.*, 101 (2014), pp. 789--827], [W. Liu and J. M. Tölle,*Electron. Commun. Probab.*, 16 (2011), pp. 447--457], [W. Liu,*J. Evol. Equations*, 9 (2009), pp. 747--770]. In particular, the results include the multivalued case of the stochastic (nonlocal) total variation flow, which solves an open problem raised in [V. Barbu, G. Da Prato, and M. Röckner,*SIAM J. Math. Anal.*, 41 (2009), pp. 1106--1120]. Moreover, under appropriate rescaling, the convergence of the unique invariant measure for the nonlocal stochastic $p$-Laplace equation to the unique invariant measure of the local stochastic $p$-Laplace equation is proven.Original language | English |
---|---|

Pages (from-to) | 4094-4125 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 48 |

Issue number | 6 |

DOIs | |

Publication status | Published - 8 Dec 2016 |

MoE publication type | A1 Journal article-refereed |

### Keywords

- Stochastic variational inequality
- Nonlocal stochastic partial differential equations
- Singular-degenerate SPDE
- stochastic $p$-Laplace equation
- ergodicity

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## Cite this

Gess, B., & Tölle, J. (2016). Ergodicity and local limits for stochastic local and nonlocal

*p*-Laplace equations.*SIAM Journal on Mathematical Analysis*,*48*(6), 4094-4125. https://doi.org/10.1137/15M1049774