Convergence of invariant measures for singular stochastic diffusion equations

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:122

Article

Ciotir, Ioana²  Toelle, Jonas M.¹ 

[1] Tech Univ Berlin, Inst Math, D-10623 Berlin, Germany

[2] Alexandru Ioan Cuza Univ, Fac Econ & Business Adm, Dept Math, Iasi, Romania

关键词: Stochastic evolution equation;    Stochastic diffusion equation;    p-Laplace equation;    I-Laplace equation;    Total variation flow;    Fast diffusion equation;    Ergodic semigroup;    Unique invariant measure;    Variational convergence;   

DOI  :  10.1016/j.spa.2011.11.011

来源: Elsevier

【 摘 要 】

It is proved that the solutions to the singular stochastic p-Laplace equation, p is an element of (1 ,2) and the solutions to the stochastic fast diffusion equation with nonlinearity parameter r is an element of (0, 1) on a bounded open domain Lambda subset of 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 L-2 (Lambda), H-1 (Lambda) 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). (c) 2011 Elsevier B.V. All rights reserved.

【 授权许可】

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