On the Relationship Between Solutions of Stochastic and Random Differential Inclusions

Caraballo, Tomás; Langa, José A.; Valero, José

doi:10.1081/sap-120020425

Cited by 5 publications

(2 citation statements)

References 10 publications

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“…The common point of view in our treatment of the stochastic problem (E β,G ) is to consider the problem as a special case of a class of abstract Cauchy problems, with an additive white noise, on the Hilbert space H. The keystone of our treatment is based on a basic idea, which under some different formulations, seems to be already quoted in [7] (see also [20], [2], [14], [27], [3] and [4]). It consists in introducing a suitable change of variables reducing the Cauchy problems to a pathwise random PDE, hence an essentially "deterministic" formulation depending on a random parameter ω ∈ Ω.…”

Section: Introductionmentioning

confidence: 99%

Stochastic energy balance climate models with Legendre weighted diffusion and a cylindrical Wiener process forcing

D́ıaz¹,

Díaz²

2021

Preprint

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We consider a class of one-dimensional nonlinear stochastic parabolic problems associated to Sellers and Budyko diffusive energy balance climate models with a Legendre weighted diffusion and an additive cylindrical Wiener processes forcing. Our results use in an important way that, under suitable assumptions on the Wiener processes, a suitable change of variables leads the problem to a pathwise random PDE, hence an essentially "deterministic" formulation depending on a random parameter. Two applications are also given: the stability of solutions when the Wiener process converges to zero and the asymptotic behaviour of solutions for large time. * Partially supported the UCM Research Group MOMAT (ref. 910480) and the projects MTM2017-85449-P and PID2020-112517GB-I00 of the DGISPI, Spain.

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Section: Introductionmentioning

confidence: 99%

Stochastic energy balance climate models with Legendre weighted diffusion and a cylindrical Wiener process forcing

D́ıaz¹,

Díaz²

2021

Preprint

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mentioning

confidence: 99%

Stochastic energy balance climate models with Legendre weighted diffusion and an additive cylindrical Wiener process forcing

Alcolea-Díaz

Díaz

2022

DCDS-S

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<p style='text-indent:20px;'>We consider a class of one-dimensional nonlinear stochastic parabolic problems associated to Sellers and Budyko diffusive energy balance climate models with a Legendre weighted diffusion and an additive cylindrical Wiener processes forcing. Our results use in an important way that, under suitable assumptions on the Wiener processes, a suitable change of variables leads the problem to a pathwise random PDE, hence an essentially "deterministic" formulation depending on a random parameter. Two applications are also given: the stability of solutions when the Wiener process converges to zero and the asymptotic behaviour of solutions for large time.</p>

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A random attractor of a stochastically perturbed evolution inclusion

Kapustyan¹

2004

Diff Equat

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After the book [1] had been published, the theory of random dynamical systems became a convenient technique for the analysis of random and stochastic differential equations. In the framework of this theory, there arises a natural notion of random attractor as a random set attracting the trajectories from −∞ [2]. This notion was generalized in [3] to the multivalued case. In the present paper, we use the theory of random attractors of multivalued random dynamical systems to analyze the qualitative behavior of the evolution equation perturbed by additive white noise:Here the multimapping F does not necessarily satisfy the Lipschitz condition.In Section 1, we refine and extend the results from [3] to problems of the form (1). The conditions obtained will be verified in Section 2. MULTIVALUED RANDOM DYNAMICAL SYSTEMSbe the set of all nonempty closed (respectively, bounded) subsets of X, let σ(X) be the Borel σ-algebra, let (Ω, Φ, P ) be a probability space, and let θ t : Ω → Ω be a measure-preserving one-parameter transformation group of Ω such that the mapping (t, w) → θ t w is (σ(R) × Φ)-measurable. Definition 1. A mapping G : R + × Ω × X → C(X) is called a multivalued random dynamical system if G is measurable [4] and the following conditions are valid for w ∈ Ω :By [4], if (Y, Φ) is a measurable space and F : Y → C(X), then F is measurable provided that F −1 (C) ∈ Φ for every closed C ⊂ X.A random set D(w) is defined as a measurable mapping D : Ω → C(X). Definition 2. A compact random set A(w)is referred to as a global random attractor (an attractor in what follows) if the following conditions are satisfied for P -almost all w ∈ Ω :Note that property (A2) characterizes attraction with probability 1 "from −∞" (pullback attraction). If property (A2) takes place, then the dynamics in "forward time" (forward attraction) is described by the following convergence in probability [2]:Suppose that we have an m-semiflow G : R + × X → C(X) describing the dynamics of a deterministic autonomous evolution equation without uniqueness [5]. Let the equation be perturbed by a stochastic additional term depending on the parameter ε ∈ [0, 1]. Let the resulting stochastic equation generate a multivalued random dynamical system G ε with attractor A ε (w). The dependence of the random attractor A ε (w) on the parameter ε is described in the following easy-to-verify assertion.

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On the Relationship Between Solutions of Stochastic and Random Differential Inclusions

Cited by 5 publications

References 10 publications

Stochastic energy balance climate models with Legendre weighted diffusion and a cylindrical Wiener process forcing

Stochastic energy balance climate models with Legendre weighted diffusion and a cylindrical Wiener process forcing

Stochastic energy balance climate models with Legendre weighted diffusion and an additive cylindrical Wiener process forcing

A random attractor of a stochastically perturbed evolution inclusion

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