Stochastic Integration of Operator-Valued Functions with Respect to Banach Space-Valued Brownian Motion

Neerven, J. M. A. M. van; Weis, Lutz

doi:10.1007/s11118-008-9088-2

Cited by 28 publications

(30 citation statements)

References 19 publications

(25 reference statements)

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“…Also, by a simple application of Fubini's theorem, {I ⊗ S(t): t 0} is R-bounded. Combining these facts with Proposition 6.5, for F ∈ L p we obtain The next proposition has been proved in [33, Theorem 3.5, Remark 3.6] (for H = C) and can be extended to a more general class of Banach spaces including the spaces L p [22,44] (in [44] a generalisation of the crucial ingredient [33, Proposition 3.3] is obtained). Proposition 7.5.…”

Section: Intermezzo Ii: H ∞ -Functional Calculimentioning

confidence: 66%

Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces

Maas

Neerven

2009

Journal of Functional Analysis

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Let (E, H, μ) be an abstract Wiener space and let D V := V D, where D denotes the Malliavin derivative and V is a closed and densely defined operator from H into another Hilbert space H . Given a bounded operator B on H , coercive on the range R(V ), we consider the operators A := V * BV in H and A := V V * B in H , as well as the realisations of the operators L := D * V BD V and L := D V D * V B in L p (E, μ) and L p (E, μ; H ) respectively, where 1 < p < ∞. Our main result asserts that the following four assertions are equivalent:) A admits a bounded H ∞ -functional calculus on R(V ). Moreover, if these conditions are satisfied, then D(L) = D(D 2 V ) ∩ D(D A ). The equivalence (1)-(4) is a nonsymmetric generalisation of the classical Meyer inequalities of Malliavin calculus (where H = H , V = I , B = 1 2 I ). A one-sided version of (1)-(4), giving L p -boundedness of the Riesz transform D V / √ L in terms of a square function estimate, is also obtained. As an application let −A generate an analytic C 0 -contraction ✩ 2411semigroup on a Hilbert space H and let −L be the L p -realisation of the generator of its second quantisation. Our results imply that two-sided bounds for the Riesz transform of L are equivalent with the Kato square root property for A. The boundedness of the Riesz transform is used to obtain an L p -domain characterisation for the operator L.

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Section: Intermezzo Ii: H ∞ -Functional Calculimentioning

confidence: 66%

Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces

Maas

Neerven

2009

Journal of Functional Analysis

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“…• B is γ-radonifying, E has property (α + ), and (SCP C ) admits a solution for all rank 1 operators C : H → E [26].…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

Invariant measures for the linear stochastic Cauchy problem and R-boundedness of the resolvent

Neerven

Weis²

2006

J. evol. equ.

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Abstract. We study the asymptotic behaviour of solutions of the stochastic abstract Cauchy problemwhere A is the generator of a C 0 -semigroup on a Banach space E, W H is a cylindrical Brownian motion over a separable Hilbert space H, and B ∈ L (H, E) is a bounded operator. Assuming the existence of a solution U , we prove that a unique invariant measure exists if the resolvent R(λ, A) is Rbounded in the right half-plane {Re λ > 0}, and that conversely the existence of an invariant measure implies the R-boundedness of R(λ, A)B in every halfplane properly contained in {Re λ > 0}. We study various abscissae related to the above problem and show, among other things, that the abscissa of Rboundedness of the resolvent of A coincides with the abscissa corresponding to the existence of invariant measures for all γ-radonifying operators B provided the latter abscissa is finite. For Hilbert spaces E this result reduces to the Gearhart-Herbst-Prüss theorem.

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“…The starting point is the choice of the space E. One of the reasons for the recent interest in UMD spaces is that they are those Banach spaces where the Wiener integral of L(U, E)-valued functions can be extended to Itô integral of L(U, E)-valued stochastic processes, see the seminal papers by van Neerven et al [28].…”

Section: Preliminaries On Stochastic Integration Theory In Banach Spacesmentioning

confidence: 99%

Volterra equations in Banach spaces with completely monotone kernels

Bonaccorsi

Desch

2012

Nonlinear Differ. Equ. Appl.

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Abstract. We consider a class of infinite delay equations in Banach spaces that models arising in the theory of viscoelasticity, for instance. The equation involves a completely monotone convolution kernel with a singularity at t = 0 and a sectorial linear spatial operator. Our main goal here is the construction of a semigroup formulation for the integral equation; in the last part of the paper, we illustrate the potentiality of the approach by considering a stochastic perturbation of the problem. Existence and uniqueness of a weak solution is established. The corresponding evolutionary solution process is Markovian, and the tools of linear analytic semigroup theory can be utilized.Mathematics Subject Classification. 60H15, 60H20, 45K05.

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Stochastic Integration of Operator-Valued Functions with Respect to Banach Space-Valued Brownian Motion

Cited by 28 publications

References 19 publications

Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces

Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces

Invariant measures for the linear stochastic Cauchy problem and R-boundedness of the resolvent

Volterra equations in Banach spaces with completely monotone kernels

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