On analyticity of Ornstein-Uhlenbeck semigroups

Beniamin Goldys

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1999)

  • Volume: 10, Issue: 3, page 131-140
  • ISSN: 1120-6330

Abstract

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Let ( R t be a transition semigroup of the Hilbert space-valued nonsymmetric Ornstein-Uhlenbeck process and let μ denote its Gaussian invariant measure. We show that the semigroup ( R t is analytic in L 2 ⁢ μ if and only if its generator is variational. In particular, we show that the transition semigroup of a finite dimensional Ornstein-Uhlenbeck process is analytic if and only if the Wiener process is nondegenerate.

How to cite

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Goldys, Beniamin. "On analyticity of Ornstein-Uhlenbeck semigroups." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 10.3 (1999): 131-140. <http://eudml.org/doc/252271>.

@article{Goldys1999,
abstract = {Let \( (R\_\{t\} \) be a transition semigroup of the Hilbert space-valued nonsymmetric Ornstein-Uhlenbeck process and let \( \mu \) denote its Gaussian invariant measure. We show that the semigroup \( (R\_\{t\} \) is analytic in \( L^\{2\} (\mu) \) if and only if its generator is variational. In particular, we show that the transition semigroup of a finite dimensional Ornstein-Uhlenbeck process is analytic if and only if the Wiener process is nondegenerate.},
author = {Goldys, Beniamin},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Ornstein-Uhlenbeck semigroup; Bilinear form; Variational generator; Polynomial chaos; Second quantization; bilinear form; variational generator; polynomial chaos; second quantization},
language = {eng},
month = {9},
number = {3},
pages = {131-140},
publisher = {Accademia Nazionale dei Lincei},
title = {On analyticity of Ornstein-Uhlenbeck semigroups},
url = {http://eudml.org/doc/252271},
volume = {10},
year = {1999},
}

TY - JOUR
AU - Goldys, Beniamin
TI - On analyticity of Ornstein-Uhlenbeck semigroups
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1999/9//
PB - Accademia Nazionale dei Lincei
VL - 10
IS - 3
SP - 131
EP - 140
AB - Let \( (R_{t} \) be a transition semigroup of the Hilbert space-valued nonsymmetric Ornstein-Uhlenbeck process and let \( \mu \) denote its Gaussian invariant measure. We show that the semigroup \( (R_{t} \) is analytic in \( L^{2} (\mu) \) if and only if its generator is variational. In particular, we show that the transition semigroup of a finite dimensional Ornstein-Uhlenbeck process is analytic if and only if the Wiener process is nondegenerate.
LA - eng
KW - Ornstein-Uhlenbeck semigroup; Bilinear form; Variational generator; Polynomial chaos; Second quantization; bilinear form; variational generator; polynomial chaos; second quantization
UR - http://eudml.org/doc/252271
ER -

References

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  2. Chojnowska-Michalik, A. - Goldys, B., Nonsymmetric Ornstein-Uhlenbeck semigroup as second quantized operator. J. Math. Kyoto Univ., 36, 1996, 481-498. Zbl0882.47013MR1417822
  3. Chojnowska, A. - Goldys, B., On regularity properties of nonsymmetric Ornstein-Uhlenbeck semigroup in L p ⁢ μ spaces. Stochastics and Stochastics Rep., 59, 1996, 183-209. Zbl0876.60039MR1427738
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  5. Da Prato, G. - Lunardi, A., On the Ornstein-Uhlenbeck operator in spaces of continuous functions. J. Funct. Anal., 131, 1995, 94-114. Zbl0846.47004MR1343161DOI10.1006/jfan.1995.1084
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  7. Da Prato, G. - Zabczyk, J., Ergodicity for infinite dimensional systems. Cambridge University Press, 1996. Zbl0849.60052MR1417491DOI10.1017/CBO9780511662829
  8. Fuhrman, M., Analyticity of transition semigroups and closability of bilinear forms in Hilbert spaces. Studia Mathematica, 115, 1995, 53-71. Zbl0830.47033MR1347432
  9. Lunardi, A., On the Ornstein-Uhlenbeck operator in L 2 ⁢ μ spaces with respect to invariant measures. Trans. Amer. Math. Soc., 349 (1997) 155-169. Zbl0890.35030MR1389786DOI10.1090/S0002-9947-97-01802-3
  10. Ma, Z. M. - Röckner, M., Dirichlet forms. Springer, 1992. 
  11. Schmuland, B., Non-symmetric Ornstein-Uhlenbeck processes in Banach space via Dirichlet forms. Can. J. Math., 45, 1993, 1324-1338. Zbl0801.31003MR1247550DOI10.4153/CJM-1993-075-6
  12. Simon, B., The P ⁢ ϕ 2 Euclidean (quantum) field theory. Zbl1175.81146

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