Ergodicity and Local Limits for Stochastic Local and Nonlocal $p$-Laplace Equations

Gess, Benjamin; Tölle, Jonas M.

doi:10.1137/15m1049774

Cited by 20 publications

(24 citation statements)

References 57 publications

(89 reference statements)

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“…The e-property has proven vital in the proof of existence and uniqueness of invariant measures for SPDE with degenerate noise (cf. [31,30,24,40]).…”

Section: Assume In Addition Thatmentioning

confidence: 99%

Well-posedness and regularity for quasilinear degenerate parabolic-hyperbolic SPDE

Gess

Hofmanová

2018

Ann. Probab.

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We study quasilinear degenerate parabolic-hyperbolic stochastic partial differential equations with general multiplicative noise within the framework of kinetic solutions. Our results are twofold: First, we establish new regularity results based on averaging techniques. Second, we prove the existence and uniqueness of solutions in a full L 1 setting requiring no growth assumptions on the nonlinearities. In addition, we prove a comparison result and an L 1 -contraction property for the solutions, generalizing the results obtained in [16].is Feller and satisfies the e-property (cf.[40]).As mentioned above, we prove new regularity estimates for kinetic solutions to (1.1) of the typefor some α > 0, based on stochastic velocity averaging lemmas. In particuar, these estimates provide a smoothing property with respect to the initial data, which typically is the first step towards the construction of an invariant measure.Even in the case of pure stochastic porous media equations (1.2) this extends previously available regularity results. For related deterministic results, see [7,23,38,52], for stochastic hyperbolic conservation laws see [17]. Our approach is mainly based on [52], but substantial difficulties due to the stochastic integral have to be overcome. Indeed, in most of the deterministic results (with the notable exception of [7]), the time variable does not play a special role and is regarded as another space variable and, in particular, space-time Fourier transforms are employed in the proofs. This changes in the stochastic case due to the irregularity of the noise in time. Therefore, it was argued

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“…The e-property has proven vital in the proof of existence and uniqueness of invariant measures for SPDE with degenerate noise (cf. [31,30,24,40]).…”

Section: Assume In Addition Thatmentioning

confidence: 99%

Well-posedness and regularity for quasilinear degenerate parabolic-hyperbolic SPDE

Gess

Hofmanová

2018

Ann. Probab.

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“…This joins the two active fields of nonlocal PDE and quasilinear SPDE, giving rise to new, intriguing questions such as convergence to local limits (cf. Section 5 below) as well as ergodicity and convergence of invariant measures of nonlocal SPDE, which is addressed in the subsequent work [25]. In the deterministic case (i.e.…”

Section: T ] × ω; H)mentioning

confidence: 99%

Stability of solutions to stochastic partial differential equations

Gess

Tölle

2016

Journal of Differential Equations

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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...

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“…A previous approach to the long-time behaviour of Markov processes stemming from monotone SPDEs with singular drift, by which the present article is inspired, is [32], which in turn uses the more abstract framework of [31]. In these works, the existence and uniqueness of invariant probability measures to stochastic local and non-local p-Laplace equations is proved, where the multivalued regime p = 1 is included.…”

Section: Introductionmentioning

confidence: 99%

Ergodicity for Singular-Degenerate Stochastic Porous Media Equations

Neuß

2021

J Dyn Diff Equat

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Ergodicity and Local Limits for Stochastic Local and Nonlocal $p$-Laplace Equations

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

Cited by 20 publications

References 57 publications

Well-posedness and regularity for quasilinear degenerate parabolic-hyperbolic SPDE

Well-posedness and regularity for quasilinear degenerate parabolic-hyperbolic SPDE

Stability of solutions to stochastic partial differential equations

Ergodicity for Singular-Degenerate Stochastic Porous Media Equations

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