Stabilization by noise for a class of stochastic reaction-diffusion equations

Cerrai, Sandra

doi:10.1007/s00440-004-0421-4

Cited by 31 publications

(20 citation statements)

References 23 publications

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“…However, the results obtained here are essentially stochastic not only because the stabilizable controller arises as multiplicative term of a Brownian N -dimensional motion but mainly because the asymptotic nature of stabilization results as well as the stochastic approach have no analogue in deterministic stabilization technique. As a matter of fact, it was known long time ago that one might use the multiplicative noise to stabilize differential systems (see [3]) and more recent results in this direction can be found in [1,2,[7][8][9]19]. (See also [11] for related results.)…”

Section: ∇)X) D(a) = D(a)mentioning

confidence: 99%

The internal stabilization by noise of the linearized Navier-Stokes equation

Barbu¹

2009

ESAIM: COCV

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Section: ∇)X) D(a) = D(a)mentioning

confidence: 99%

The internal stabilization by noise of the linearized Navier-Stokes equation

Barbu¹

2009

ESAIM: COCV

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“…In the case of the stochastic reaction diffusion equation, equation (1.1) is mostly considered in a suitable chosen Banach space of continuous functions, see [10,6,7,8,14]. It is our main goal in this work to make optimal use of the regularity of the linear part of (1.1).…”

Section: Introductionmentioning

confidence: 99%

“…8,2008 By assumption, the right hand side converges to zero as N, M tend to infinity, so that W N A (t)(ω) converges in V γ locally uniformly. The limit W A (t)(ω) is therefore continuous on V γ .…”

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confidence: 99%

Invariant measures for semilinear SPDE’s with local Lipschitz drift coefficients and applications

Es-Sarhir

Stannat

2007

J. evol. equ.

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“…(see [12] for a proof and see also [5] for an analogous result for equations with non-Lipschitz coefficients). Therefore, as…”

Section: )mentioning

confidence: 99%

A basic identity for Kolmogorov operators in the space of continuous functions related to RDEs with multiplicative noise

Cerrai¹,

Prato²

2014

Ann. Probab.

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We consider the Kolmogorov operator associated with a reaction-diffusion equation having polynomially growing reaction coefficient and perturbed by a noise of multiplicative type, in the Banach space E of continuous functions. By analyzing the smoothing properties of the associated transition semigroup, we prove a modification of the classical identité du carré di champs that applies to the present non-Hilbertian setting. As an application of this identity, we construct the Sobolev space W 1,2 (E; µ), where µ is an invariant measure for the system, and we prove the validity of the Poincaré inequality and of the spectral gap. * Key words and phrases: Stochastic reaction-diffusion equations, Kolmogorov operators, Poincaré inequality, spectral gap, Sobolev spaces in infinite dimensional spaces . † Partially supported by the NSF grant DMS0907295 "Asymptotic Problems for SPDE's".and this allowed us to prove that the series in (1.5) is convergent, for any ϕ ∈ D(K). To this purpose, we would like to mention the fact that our duality argument does work because we

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Stabilization by noise for a class of stochastic reaction-diffusion equations

Cited by 31 publications

References 23 publications

The internal stabilization by noise of the linearized Navier-Stokes equation

The internal stabilization by noise of the linearized Navier-Stokes equation

Invariant measures for semilinear SPDE’s with local Lipschitz drift coefficients and applications

A basic identity for Kolmogorov operators in the space of continuous functions related to RDEs with multiplicative noise

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