Singular-Degenerate Multivalued Stochastic Fast Diffusion Equations

Gess, Benjamin; Röckner, Michael

doi:10.1137/151003726

Cited by 21 publications

(25 citation statements)

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“…By a slight modification of [24,Appendix C] this solution is also a continuous SVI solution to (6.14). We further note that by [23] for ψ(r) = Let X n be the unique (limit) solutions to (6.14) with ψ replaced by ψ n and X be the unique SVI solution to (6.14) with ψ as above. Then…”

Section: Trotter Type Resultsmentioning

confidence: 99%

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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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Section: Trotter Type Resultsmentioning

confidence: 99%

“…Definition 2.1 modifies notions of stochastic SVI solutions introduced in [11,13,22,23]. These modifications are chosen in order to obtain a stable notion of solutions with regard to approximations of the subdifferential ϕ.…”

Section: Generalities On Stochastic Variational Inequalitiesmentioning

confidence: 99%

Stability of solutions to stochastic partial differential equations

Gess

Tölle

2016

Journal of Differential Equations

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“…Uniqueness. Compare with [10,17,26,27,30]. Let X ∈ L 2 ([0, T ] × Ω; H) be any SVI solution to (2.2) with initial datum x ∈ L 2 (Ω, F 0 , P; H).…”

Section: 4mentioning

confidence: 99%

“…[10]. Among others, SVI-solutions to stochastic evolution equations have also been discussed in [12,26,27,29,45]. As seen in [30], the SVI-approach is quite robust under perturbations of the convex-subpotential-type drift with respect to the Mosco-topology.…”

Section: Introductionmentioning

confidence: 99%

Stochastic evolution equations with singular drift and gradient noise via curvature and commutation conditions

Tölle

2020

Stochastic Processes and their Applications

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We prove existence and uniqueness of solutions to a nonlinear stochastic evolution equation on the d-dimensional torus with singular p-Laplace-type or total variation flowtype drift with general sublinear doubling nonlinearities and Gaussian gradient Stratonovich noise with divergence-free coefficients. Assuming a weak defective commutator bound and a curvature-dimension condition, the well-posedness result is obtained in a stochastic variational inequality setup by using resolvent and Dirichlet form methods and an approximative Itô-formula.

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“…The fact that (3.13) has a unique H −1 m+1 -solution follows from Theorem 3.3. For the remaining properties, let us consider the approximation 15) where N depends only on K, T, d, q and ε.…”

Section: (312)mentioning

confidence: 99%