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Gauss-Markov processes on Hilbert spaces


Authors: Ben Goldys, Szymon Peszat and Jerzy Zabczyk
Journal: Trans. Amer. Math. Soc. 368 (2016), 89-108
MSC (2010): Primary 60G15, 60H15, 60J99
DOI: https://doi.org/10.1090/tran/6329
Published electronically: April 3, 2015
MathSciNet review: 3413857
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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

DOI: https://doi.org/10.1090/tran/6329
Keywords: Gauss--Markov process, Ornstein--Uhlenbeck process, Gaussian measure, bw-topology, strict topology, generator
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.
Article copyright: © Copyright 2015 American Mathematical Society