We provide an abstract variational existence and uniqueness result for multi-valued, monotone, non-coercive stochastic evolution inclusions in Hilbert spaces with general additive and Wiener multiplicative noise. As examples we discuss certain singular diffusion equations such as the stochastic 1-Laplacian evolution (total variation flow) in all space dimensions and the stochastic singular fast diffusion equation. In case of additive Wiener noise we prove the existence of a unique weak-* mean ergodic invariant measure.Comment: 39 pages, in press: J. Math. Pures Appl. (2013

Abstract. We provide a general framework for the stability of solutions to stochastic partial differential equations with respect to perturbations of the drift. More precisely, we consider stochastic partial differential equations with drift given as the subdifferential of a convex function and prove continuous dependence of the solutions with regard to random Mosco convergence of the convex potentials. In particular, we identify the concept of stochastic variational inequalities (SVI) as a well-suited framework to study such stability properties. The generality of the developed framework is then laid out by deducing Trotter type and homogenization results for stochastic fast diffusion and stochastic singular p-Laplace equations. In addition, we provide an SVI treatment for stochastic nonlocal p-Laplace equations and prove their convergence to the respective local models. IntroductionWe consider the stability of stochastic partial differential equations of the general type with respect to perturbations of the convex, lower-semicontinuous potential ϕ, defined on some separable Hilbert space H. Here, W is a cylindrical Wiener process on a separable Hilbert space U and B : H → L 2 (U, H) are Lipschitz continuous diffusion coefficients. We are especially interested in applications to quasilinear, singular-degenerate SPDE, such as the stochastic singular p-Laplace equationwith p ∈ [1, 2), which will serve as a model example in the introduction. In particular, this generalizes results obtained in [11,13,24] on the multi-valued case of the stochastic total variation flow (p = 1). In the deterministic case, i.e. B ≡ 0 in (1.1), the stability of solutions with respect to ϕ is well-understood [6]. More precisely, for a sequence ϕ n of convex, lowersemicontinuous functions on H and corresponding solutions X n it is known that the convergence of ϕ n to ϕ in Mosco sense (cf. Appendix B below) implies the convergence of X n to X.In the stochastic case (1.1) much less is known and only particular examples could be treated so far [10,[17][18][19][20] (cf. Section 1.1 below). In particular, the singular nature of (1.2) and the resulting low regularity of the solutions lead to difficulties in proving stability with respect to perturbations of the drift ∂ϕ. In this work we introduce the notion of random Mosco convergence of convex, lower-semicontinuous functionals ϕ n and prove that if ϕ n → ϕ in random Mosco sense, then the corresponding solutions X n to (1.1) converge weakly, that is,A key ingredient of the proof of this result is the right choice of a notion of a solution to (1.1). Due to the low regularity of solutions to singular SPDE such as (1.2) (especially for p = 1), an appropriate notion of a solution needs to rely on little regularity only. We identify the SVI approach to SPDE to be a well-suited framework to study stability questions for SPDE of the type (1.1). The abstract convergence results are then applied to a variety of examples, that become immediate consequences of the abstract theory. For the sake of the int...

Abstract. Ergodicity for local and nonlocal stochastic singular p-Laplace equations is proven, without restriction on the spatial dimension and for all

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 ⊂ Ê d withDirichlet boundary conditions are continuous in mean, uniformly in time, with respect to the parameters p and r respectively (in the Hilbert spaces L 2 (Λ), H −1 (Λ) respectively). The highly singular limit case p = 1 is treated with the help of stochastic evolution variational inequalities, where È-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).

We define compositions ϕ(X) of Hölder paths X in R n and functions of bounded variation ϕ under a relative condition involving the path and the gradient measure of ϕ. We show the existence and properties of generalized Lebesgue-Stieltjes integrals of compositions ϕ(X) with respect to a given Hölder path Y . These results are then used, together with Doss' transform, to obtain existence and, in a certain sense, uniqueness results for differential equations in R n driven by Hölder paths and involving coefficients of bounded variation. Examples include equations with discontinuous coefficients driven by paths of two-dimensional fractional Brownian motions.