The divergence of Banach space valued random variables on Wiener space

Mayer-Wolf, Eddy; Zakai, Moshe

doi:10.1007/s00440-004-0397-0

Cited by 7 publications

(6 citation statements)

References 23 publications

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“…We end this section considering a field b taking values outside H, to which our theory applies (although well-posedness was already shown in [MWZ05]). Assume that that each eigenvalue of Q admits a two-dimensional eigenspace thus, slightly changing the notation, we write (e i ,ẽ i ) for an orthonormal basis of H consisting of eigenvectors of Q.…”

Section: Gaussian Hilbert Spacesmentioning

confidence: 95%

Well-posedness of Lagrangian flows and continuity equations in metric measure spaces

2014

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We establish, in a rather general setting, an analogue of DiPerna-Lions theory on wellposedness of flows of ODE's associated to Sobolev vector fields. Key results are a wellposedness result for the continuity equation associated to suitably defined Sobolev vector fields, via a commutator estimate, and an abstract superposition principle in (possibly extended) metric measure spaces, via an embedding into R ∞ . When specialized to the setting of Euclidean or infinite dimensional (e.g. Gaussian) spaces, large parts of previously known results are recovered at once. Moreover, the class of RCD(K, ∞) metric measure spaces introduced in [AGS14b] and object of extensive recent research fits into our framework. Therefore we provide, for the first time, wellposedness results for ODE's under low regularity assumptions on the velocity and in a nonsmooth context.

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Section: Gaussian Hilbert Spacesmentioning

confidence: 95%

Well-posedness of Lagrangian flows and continuity equations in metric measure spaces

2014

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“…(ii) If ∈ dom p and y ∈ Y , it follows directly from the definitions that ⊗ y ∈ dom p,Y and that ( ⊗ y) = ( )y. [9,Proposition 3.14]). An element K ∈ L p ( ; L (W * , Y )) belongs to dom p,Y if and only if K T l ∈ dom p for every l ∈ Y * and for some C >0…”

Section: Stochastic Analysis Preliminariesmentioning

confidence: 97%

“…In the scalar case this abstract Wiener space version of the Clark-Ocone formula has already been considered in [12,16,19]. Section 2 is devoted to some basic notions of stochastic analysis in Wiener space, including the gradient and divergence operators, the latter applied to random variables which are not necessarily H-valued, as introduced in [9]. It should be noted that the tangent processes considered in [1] and [3] are examples of such random vectors.…”

Section: Introductionmentioning

confidence: 99%

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The Clark–Ocone formula for vector valued Wiener functionals

Mayer-Wolf¹,

Zakai²

2005

Journal of Functional Analysis

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The classical representation of random variables as the Itô integral of nonanticipative integrands is extended to include Banach space valued random variables on an abstract Wiener space equipped with a filtration induced by a resolution of the identity on the Cameron-Martin space. The Itô integral is replaced in this case by an extension of the divergence to random operators, and the operators involved in the representation are adapted with respect to this filtration in a suitably defined sense.A complete characterization of measure preserving transformations in Wiener space is presented as an application of this generalized Clark-Ocone formula.

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“…Vector-valued Malliavin calculus has been consider by several authors [18][19][20]33]. The main focus in this work is on the interplay between Malliavin calculus and decoupling inequalities.…”

Section: Introductionmentioning

confidence: 99%

Malliavin calculus and decoupling inequalities in Banach spaces

Maas

2010

Journal of Mathematical Analysis and Applications

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We develop a theory of Malliavin calculus for Banach space-valued random variables. Using radonifying operators instead of symmetric tensor products we extend the Wiener-Itô isometry to Banach spaces. In the white noise case we obtain two sided L p -estimates for multiple stochastic integrals in arbitrary Banach spaces. It is shown that the Malliavin derivative is bounded on vector-valued Wiener-Itô chaoses. Our main tools are decoupling inequalities for vector-valued random variables. In the opposite direction we use Meyer's inequalities to give a new proof of a decoupling result for Gaussian chaoses in UMD Banach spaces.

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The divergence of Banach space valued random variables on Wiener space

Cited by 7 publications

References 23 publications

Well-posedness of Lagrangian flows and continuity equations in metric measure spaces

Well-posedness of Lagrangian flows and continuity equations in metric measure spaces

The Clark–Ocone formula for vector valued Wiener functionals

Malliavin calculus and decoupling inequalities in Banach spaces

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