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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles 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 1.48.

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On the local time process of a skew Brownian motion
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by Andrei Borodin and Paavo Salminen PDF
Trans. Amer. Math. Soc. 372 (2019), 3597-3618 Request permission

Abstract:

We derive a Ray–Knight type theorem for the local time process (in the space variable) of a skew Brownian motion up to an independent exponential time. It is known that the local time seen as a density of the occupation measure and taken with respect to the Lebesgue measure has a discontinuity at the skew point (in our case at zero), but the local time taken with respect to the speed measure is continuous. In this paper we discuss this discrepancy by characterizing the dynamics of the local time process in both of these cases. The Ray–Knight type theorem is applied to study integral functionals of the local time process of the skew Brownian motion. In particular, we determine the distribution of the maximum of the local time process up to a fixed time, which can be seen as the main new result of the paper.
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Additional Information
  • Andrei Borodin
  • Affiliation: National Research University, Higher School of Economics, Campus Saint Petersburg, Saint Petersburg, Russia
  • MR Author ID: 190985
  • Email: borodin@pdmi.ras.ru
  • Paavo Salminen
  • Affiliation: Faculty of Science and Engineering, Abo Akademi, Turku, Finland
  • MR Author ID: 153560
  • Email: paavo.salminen@abo.fi
  • Received by editor(s): October 18, 2018
  • Received by editor(s) in revised form: February 5, 2019
  • Published electronically: June 3, 2019
  • Additional Notes: This research was partially supported by a grant from Magnus Ehrnrooths stiftelse, Finland, and grant SPbU-DGF 6.65.37.2017.
  • © Copyright 2019 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 372 (2019), 3597-3618
  • MSC (2010): Primary 60J65, 60J60, 60J55
  • DOI: https://doi.org/10.1090/tran/7852
  • MathSciNet review: 3988620