On analytic Ornstein-Uhlenbeck semigroups in infinite dimensions

Maas, Jan; Neerven, Jan van

doi:10.1007/s00013-007-2082-x

Cited by 17 publications

(38 citation statements)

References 10 publications

(13 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…2 Remark 4.6. We mention that a different argument to establish analyticity in L p has been given in the case of Ornstein-Uhlenbeck semigroups in [10,37]. This argument also works in the more general setting considered here and yields an angle of analyticity which is better than the one obtained by Stein interpolation.…”

Section: The Operator Lmentioning

confidence: 86%

“…This argument also works in the more general setting considered here and yields an angle of analyticity which is better than the one obtained by Stein interpolation. For Ornstein-Uhlenbeck semigroups (for which we have B + B * = I , see [37]), this angle is optimal. Definition 4.7.…”

Section: The Operator Lmentioning

confidence: 97%

“…It has been shown in [37] that if P is analytic for some (equivalently, for all) 1 < p < ∞, then its infinitesimal generator −L is of the form considered in Section 2. More precisely, there exists a unique coercive operator B on H such that…”

Section: Linear Stochastic Evolution Equationsmentioning

confidence: 98%

See 2 more Smart Citations

Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces

Maas

Neerven

2009

Journal of Functional Analysis

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: The Operator Lmentioning

confidence: 86%

Section: The Operator Lmentioning

confidence: 97%

See 1 more Smart Citation

Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces

Maas

Neerven

2009

Journal of Functional Analysis

Self Cite

View full text Add to dashboard Cite

show abstract

“…This characterization is the starting point of [21], where the authors generalize the results in [6] to the infinite dimensional case. To begin with, they prove that the operator B ∈ L(H), which is the extension of Q ∞ A * to the whole H, satisfies…”

Section: R Nmentioning

confidence: 91%

Analyticity of Nonsymmetric Ornstein-Uhlenbeck Semigroup with Respect to a Weighted Gaussian Measure

Addona

2020

Potential Anal

View full text Add to dashboard Cite

In this paper we show that the realization in L p (X, ν∞) of a nonsymmetric Ornstein-Uhlenbeck operator Lp is sectorial for any p ∈ (1, +∞) and we provide an explicit sector of analyticity. Here, (X, µ∞, H∞) is an abstract Wiener space, i.e., X is a separable Banach space, µ∞ is a centred non degenerate Gaussian measure on X and H∞ is the associated Cameron-Martin space. Further, ν∞ is a weighted Gaussian measure, that is, ν∞ = e −U µ∞ where U is a convex function which satisfies some minimal conditions. Our results strongly rely on the theory of nonsymmetric Dirichlet forms and on the divergence form of the realization of L2 in L 2 (X, ν∞).

show abstract

“…For example, there are if and only if conditions for its compactness, self-adjointness, analyticity, and hypercontractivity (see e.g. [7][8][9][10][11]14,13,[23][24][25][26]). In particular, in the case of E = R, (P t ) is symmetric, compact (even nuclear) and hypercontractiv.…”

Section: Introductionmentioning

confidence: 99%

Lévy–Ornstein–Uhlenbeck transition semigroup as second quantized operator

Peszat

2011

Journal of Functional Analysis

View full text Add to dashboard Cite

Let μ be an invariant measure for the transition semigroup (P t ) of the Markov family defined by the Ornstein-Uhlenbeck type equation dX = AX dt + dL on a Hilbert space E, driven by a Lévy process L. It is shown that for any t 0, P t considered on L 2 (μ) is a second quantized operator on a Poisson Fock space of e At . From this representation it follows that the transition semigroup corresponding to the equation on E = R, driven by an α-stable noise L, α ∈ (0, 2), is neither compact nor symmetric.

show abstract

On analytic Ornstein-Uhlenbeck semigroups in infinite dimensions

Cited by 17 publications

References 10 publications

Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces

Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces

Analyticity of Nonsymmetric Ornstein-Uhlenbeck Semigroup with Respect to a Weighted Gaussian Measure

Lévy–Ornstein–Uhlenbeck transition semigroup as second quantized operator

Contact Info

Product

Resources

About