| 1. |
- Andersson, Adam, 1979, et al.
(författare)
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Ornstein-Uhlenbeck theory in finite dimension
- 2012
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- The contents of these notes were presented during ten lectures, in November 2011, by Peter Sjögren in Gothenburg. The text was written by Adam Andersson who participated and is improved after the careful reading by Peter Sjögren. Ornstein-Uhlenbeck theory can be described as a model of harmonic analysis in which Lebesgue measure is everywhere replaced by a Gaussian measure. The theory has applications in quantum physics and probability theory. If one passes to infinite dimensions and places the theory in a probabilistic context, one gets the Malliavin calculus. In Chapter 1, the basic theory is developed. This concerns the Hermite polynomials, the Ornstein-Uhlenbeck operator and most importantly its semigroup. The Hermite polynomials form an orthogonal system with respect to the Gaussian measure in Euclidean space. It turns out that they are the eigenfunctions of the Ornstein-Uhlenbeck operator, and since this operator is self-adjoint and positive semidefinite, the semigroup can be defined spectrally. An explicit kernel is derived for the semigroup, known as the Mehler kernel. It will be of central importance in this text. In Chapter 2, boundary convergence for the semigroup is considered, i.e., the limiting behavior of the semigroup as the “time” tends to zero. This is done by introducing a maximal operator for the semigroup and proving that it is of weak type (1,1). This result implies almost everywhere convergence for integrable boundary functions. In Chapter 3, first-order Riesz operators related to the Ornstein- Uhlenbeck operator are treated. Explicit off-diagonal kernels for these operators are found. It is finally proved that the Riesz operators are of weak type (1,1).
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| 2. |
- Casarino, V., et al.
(författare)
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On the maximal operator of a general Ornstein-Uhlenbeck semigroup
- 2022
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Ingår i: Mathematische Zeitschrift. - : Springer Science and Business Media LLC. - 0025-5874 .- 1432-1823. ; 301, s. 2393-2413
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Tidskriftsartikel (refereegranskat)abstract
- If Q is a real, symmetric and positive definite n x n matrix, and B a real n x n matrix whose eigenvalues have negative real parts, we consider the Ornstein-Uhlenbeck semigroup on R-n with covariance Q and drift matrix B. Our main result says that the associated maximal operator is of weak type (1, 1) with respect to the invariant measure. The proof has a geometric gist and hinges on the "forbidden zones method" previously introduced by the third author.
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| 3. |
- Casarino, V., et al.
(författare)
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On the orthogonality of generalized eigenspaces for the Ornstein-Uhlenbeck operator
- 2021
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Ingår i: Archiv der Mathematik. - : Springer Science and Business Media LLC. - 0003-889X .- 1420-8938. ; 117, s. 547-556
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Tidskriftsartikel (refereegranskat)abstract
- We study the orthogonality of the generalized eigenspaces of an Ornstein-Uhlenbeck operator L in R-N, with drift given by a real matrix B whose eigenvalues have negative real parts. If B has only one eigenvalue, we prove that any two distinct generalized eigenspaces of L are orthogonal with respect to the invariant Gaussian measure. Then we show by means of two examples that if B admits distinct eigenvalues, the generalized eigenspaces of L may or may not be orthogonal.
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| 4. |
- Casarino, V., et al.
(författare)
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On the variation operator for the Ornstein-Uhlenbeck semigroup in dimension one
- 2024
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Ingår i: Annali Di Matematica Pura Ed Applicata. - 0373-3114 .- 1618-1891. ; 203:1, s. 205-219
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Tidskriftsartikel (refereegranskat)abstract
- Consider the variation seminorm of the Ornstein-Uhlenbeck semigroup Ht in dimension one, taken with respect to t. We show that this seminorm defines an operator of weak type (1, 1) for the relevant Gaussian measure. The analogous Lp estimates for 1 < p < 8 were already known.
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| 5. |
- Casarino, Valentina, et al.
(författare)
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The maximal operator of a normal Ornstein--Uhlenbeck semigroup is of weak type (1,1)
- 2020
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Ingår i: Annali della Scuola normale superiore di Pisa - Classe di scienze. - : Scuola Normale Superiore - Edizioni della Normale. - 2036-2145 .- 0391-173X. ; 21, s. 385-410
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Tidskriftsartikel (refereegranskat)abstract
- Consider a normal Ornstein--Uhlenbeck semigroup in Rn, whose covariance is given by a positive definite matrix.The drift matrix is assumed to have eigenvalues only in the left half-plane.We prove that the associated maximal operatoris of weak type (1,1) with respect to the invariant measure.This extends earlier work by G. Mauceri and L. Noselli. The proof goes via the special case where thematrix defining the covariance isIand the drift matrix is diagonal.
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| 6. |
- Criado, Alberto, et al.
(författare)
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Bounds for maximal functions associated with rotational invariant measures in high dimensions
- 2014
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Ingår i: Journal of Geometric Analysis. - : Springer Science and Business Media LLC. - 1050-6926 .- 1559-002X. ; 24:2, s. 595-612
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Tidskriftsartikel (refereegranskat)abstract
- ABSTRACT. In recent articles, it was proved that when mu is a finite, radial measure in Rn with a bounded, radially decreasing density, the Lp(mu) norm of the associated maximal operator grows to infinity with the dimension for a small range of values of p near 1. We prove that when mu is Lebesgue measure restricted to the unit ball and p < 2, the Lp operator norms of the maximal operator are unbounded in dimension, even when the action is restricted to radially decreasing functions. In spite of this, this maximal operator admits dimension-free Lp bounds for every p > 2, when restricted to radially decreasing functions. On the other hand, when mu is the Gaussian measure, the Lp operator norms of the maximal operator grow to infinity with the dimension for any finite p > 1, even in the subspace of radially decreasing functions.
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