Convergence of invariant measures for singular stochastic diffusion equations

Ioana Ciotir; Jonas M. Tölle

doi:10.1016/j.spa.2011.11.011

Convergence of invariant measures for singular stochastic diffusion equations

Ioana Ciotir, Jonas M. Tölle^*

^*Corresponding author for this work

Not published at Aalto University

Research output: Contribution to journal › Article › Scientific › peer-review

12 Citations (Scopus)

Abstract

It is proved that the solutions to the singular stochastic p-Laplace equation, p∈(1,2) and the solutions to the stochastic fast diffusion equation with nonlinearity parameter r∈(0,1) on a bounded open domain Λ⊂R^d with Dirichlet boundary conditions are continuous in mean, uniformly in time, with respect to the parameters p and r respectively (in the Hilbert spaces L²(Λ), H^-1(Λ) respectively). The highly singular limit case p=1 is treated with the help of stochastic evolution variational inequalities, where P-a.s. convergence, uniformly in time, is established. It is shown that the associated unique invariant measures of the ergodic semigroups converge in the weak sense (of probability measures).

Original language	English
Pages (from-to)	1998-2017
Number of pages	20
Journal	Stochastic Processes and their Applications
Volume	122
Issue number	4
DOIs	https://doi.org/10.1016/j.spa.2011.11.011
Publication status	Published - Apr 2012
MoE publication type	A1 Journal article-refereed

Keywords

1-Laplace equation
Ergodic semigroup
Fast diffusion equation
p-Laplace equation
Stochastic diffusion equation
Stochastic evolution equation
Total variation flow
Unique invariant measure
Variational convergence

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10.1016/j.spa.2011.11.011Licence: Other

https://arxiv.org/pdf/1103.0229.pdfLicence: Other

Cite this

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title = "Convergence of invariant measures for singular stochastic diffusion equations",

abstract = "It is proved that the solutions to the singular stochastic p-Laplace equation, p∈(1,2) and the solutions to the stochastic fast diffusion equation with nonlinearity parameter r∈(0,1) on a bounded open domain Λ⊂Rd with Dirichlet boundary conditions are continuous in mean, uniformly in time, with respect to the parameters p and r respectively (in the Hilbert spaces L2(Λ), H-1(Λ) respectively). The highly singular limit case p=1 is treated with the help of stochastic evolution variational inequalities, where P-a.s. convergence, uniformly in time, is established. It is shown that the associated unique invariant measures of the ergodic semigroups converge in the weak sense (of probability measures).",

keywords = "1-Laplace equation, Ergodic semigroup, Fast diffusion equation, p-Laplace equation, Stochastic diffusion equation, Stochastic evolution equation, Total variation flow, Unique invariant measure, Variational convergence",

author = "Ioana Ciotir and T{\"o}lle, {Jonas M.}",

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language = "English",

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TY - JOUR

T1 - Convergence of invariant measures for singular stochastic diffusion equations

AU - Ciotir, Ioana

AU - Tölle, Jonas M.

PY - 2012/4

Y1 - 2012/4

N2 - It is proved that the solutions to the singular stochastic p-Laplace equation, p∈(1,2) and the solutions to the stochastic fast diffusion equation with nonlinearity parameter r∈(0,1) on a bounded open domain Λ⊂Rd with Dirichlet boundary conditions are continuous in mean, uniformly in time, with respect to the parameters p and r respectively (in the Hilbert spaces L2(Λ), H-1(Λ) respectively). The highly singular limit case p=1 is treated with the help of stochastic evolution variational inequalities, where P-a.s. convergence, uniformly in time, is established. It is shown that the associated unique invariant measures of the ergodic semigroups converge in the weak sense (of probability measures).

AB - It is proved that the solutions to the singular stochastic p-Laplace equation, p∈(1,2) and the solutions to the stochastic fast diffusion equation with nonlinearity parameter r∈(0,1) on a bounded open domain Λ⊂Rd with Dirichlet boundary conditions are continuous in mean, uniformly in time, with respect to the parameters p and r respectively (in the Hilbert spaces L2(Λ), H-1(Λ) respectively). The highly singular limit case p=1 is treated with the help of stochastic evolution variational inequalities, where P-a.s. convergence, uniformly in time, is established. It is shown that the associated unique invariant measures of the ergodic semigroups converge in the weak sense (of probability measures).

KW - 1-Laplace equation

KW - Ergodic semigroup

KW - Fast diffusion equation

KW - p-Laplace equation

KW - Stochastic diffusion equation

KW - Stochastic evolution equation

KW - Total variation flow

KW - Unique invariant measure

KW - Variational convergence

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U2 - 10.1016/j.spa.2011.11.011

DO - 10.1016/j.spa.2011.11.011

M3 - Article

AN - SCOPUS:84860224374

SN - 0304-4149

VL - 122

SP - 1998

EP - 2017

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

IS - 4

ER -

Convergence of invariant measures for singular stochastic diffusion equations

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