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 |
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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