A class of Gaussian processes with fractional spectral measures

Alpay, Daniel; Jørgensen, Palle E. T.; Levanony, David

doi:10.1016/j.jfa.2011.03.012

Cited by 46 publications

(31 citation statements)

References 29 publications

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“…Moreover, the operator T m is continuous from S (and hence from S R ) into L 2 (R). This was proved in [3], and we repeat the argument for completeness. As in [3],…”

Section: The M Noise Spacementioning

confidence: 55%

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A generalized white noise space approach to stochastic integration for a class of Gaussian stationary increment processes

Alpay¹,

Kipnis²

2013

OpMath

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Abstract. Given a Gaussian stationary increment processes, we show that a Skorokhod-Hitsuda stochastic integral with respect to this process, which obeys the Wick-Itô calculus rules, can be naturally defined using ideas taken from Hida's white noise space theory. We use the Bochner-Minlos theorem to associate a probability space to the process, and define the counterpart of the S-transform in this space. We then use this transform to define the stochastic integral and prove an associated Itô formula.

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“…Moreover, the operator T m is continuous from S (and hence from S R ) into L 2 (R). This was proved in [3], and we repeat the argument for completeness. As in [3],…”

Section: The M Noise Spacementioning

confidence: 55%

“…It has been proved in [3] that C m is continuous from S into R. Restricting C m to real-valued functions and using the Bochner-Minlos theorem, we obtain an analog of the white noise space in which the process B m is built in a natural way. Stochastic calculus with respect to this process is then developed using an S-transform approach.…”

Section: Introductionmentioning

confidence: 99%

A generalized white noise space approach to stochastic integration for a class of Gaussian stationary increment processes

Alpay¹,

Kipnis²

2013

OpMath

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“…The literature on Gaussian Hilbert space, white noise analysis, and its relevance to Malliavin calculus is vast; we limit ourselves here to citing [17,[36][37][38][39][40][41], and the papers cited there.…”

Section: Gaussian Hilbert Spacementioning

confidence: 99%

“…We begin with the following: Lemma 3.14 (Leibniz-Malliavin). Let H 1 T − → H 2 be the Malliavin derivative from Equations (36) and (37). Then,…”

Section: A Derivation On the Algebra Dmentioning

confidence: 99%

Infinite-dimensional Lie Algebras, Representations, Hermitian Duality and the Operators of Stochastic Calculus

Jørgensen

Tian

2016

Axioms

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Abstract:We study densely defined unbounded operators acting between different Hilbert spaces. For these, we introduce a notion of symmetric (closable) pairs of operators. The purpose of our paper is to give applications to selected themes at the cross road of operator commutation relations and stochastic calculus. We study a family of representations of the canonical commutation relations (CCR)-algebra (an infinite number of degrees of freedom), which we call admissible. The family of admissible representations includes the Fock-vacuum representation. We show that, to every admissible representation, there is an associated Gaussian stochastic calculus, and we point out that the case of the Fock-vacuum CCR-representation in a natural way yields the operators of Malliavin calculus. We thus get the operators of Malliavin's calculus of variation from a more algebraic approach than is common. We further obtain explicit and natural formulas, and rules, for the operators of stochastic calculus. Our approach makes use of a notion of symmetric (closable) pairs of operators. The Fock-vacuum representation yields a maximal symmetric pair. This duality viewpoint has the further advantage that issues with unbounded operators and dense domains can be resolved much easier than what is possible with alternative tools. With the use of CCR representation theory, we also obtain, as a byproduct, a number of new results in multi-variable operator theory which we feel are of independent interest.

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“…Readers are referred to the following papers, and the references cited there, for details regarding this: [25][26][27][28][29][30][31][32][33][34][35].…”

Section: Introductionmentioning

confidence: 99%

Graph Laplacians and discrete reproducing kernel Hilbert spaces from restrictions

Jørgensen

Tian

2016

Stochastic Analysis and Applications

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We study kernel functions, and associated reproducing kernel Hilbert spaces H over infinite, discrete, and countable sets V . Numerical analysis builds discrete models (e.g., finite element) for the purpose of finding approximate solutions to boundary value problems; using multiresolution-subdivision schemes in continuous domains. In this article, we turn the tables: Our object of study is realistic infinite discrete models in their own right; and we then use an analysis of suitable continuous counterpart problems, but now serving as a tool for obtaining solutions in the discrete world.

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A class of Gaussian processes with fractional spectral measures

Cited by 46 publications

References 29 publications

A generalized white noise space approach to stochastic integration for a class of Gaussian stationary increment processes

A generalized white noise space approach to stochastic integration for a class of Gaussian stationary increment processes

Infinite-dimensional Lie Algebras, Representations, Hermitian Duality and the Operators of Stochastic Calculus

Graph Laplacians and discrete reproducing kernel Hilbert spaces from restrictions

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