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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Gauss-Markov processes on Hilbert spaces
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by Ben Goldys, Szymon Peszat and Jerzy Zabczyk PDF
Trans. Amer. Math. Soc. 368 (2016), 89-108 Request permission

Abstract:

K. Itô characterised in 1984 zero-mean stationary Gauss–Markov processes evolving on a class of infinite-dimensional spaces. In this work we extend the work of Itô in the case of Hilbert spaces: Gauss–Markov families that are time-homogenous are identified as solutions to linear stochastic differential equations with singular coefficients. Choosing an appropriate locally convex topology on the space of weakly sequentially continuous functions we also characterize the transition semigroup, the generator and its core, thus providing an infinite-dimensional extension of the classical result of Courrège in the case of Gauss–Markov semigroups.
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Additional Information
  • Ben Goldys
  • Affiliation: School of Mathematics and Statistics, The University of Sydney, Sydney 2006, Australia
  • Email: Beniamin.Goldys@sydney.edu.au
  • Szymon Peszat
  • Affiliation: Institute of Mathematics, Polish Academy of Sciences, Św. Tomasza 30/7, 31-027 Cracow, Poland
  • Address at time of publication: Institute of Mathematics, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland
  • Email: napeszat@cyf-kr.edu.pl
  • Jerzy Zabczyk
  • Affiliation: Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-950 Warsaw, Poland
  • Email: J.Zabczyk@impan.pl
  • Received by editor(s): July 9, 2013
  • Received by editor(s) in revised form: October 19, 2013
  • Published electronically: April 3, 2015
  • Additional Notes: The work of the first author was partially supported by the ARC Discovery Grant DP120101886. Part of this work was prepared during his visit to the Institute of Mathematics of the Polish Academy of Sciences. He gratefully acknowledges the excellent working conditions and stimulating atmosphere of the Institute.
    The work of the second and third authors was supported by Polish National Science Center grant DEC2013/09/B/ST1/03658.
  • © Copyright 2015 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 368 (2016), 89-108
  • MSC (2010): Primary 60G15, 60H15, 60J99
  • DOI: https://doi.org/10.1090/tran/6329
  • MathSciNet review: 3413857