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Zapiski Nauchnykh Seminarov LOMI, 1985, Volume 142, Pages 6–24 (Mi znsl4224)  

This article is cited in 2 scientific papers (total in 2 papers)

Distribution of the supremum of increments of Brownian local time

A. N. Borodin
Full-text PDF (713 kB) Citations (2)
Abstract: The joint distribution of the variables $\hat t(t,r)$, $\hat t(t,r)$ and $\sup_{0\le s\le t}(\hat t(s,q)-\hat t(s,r))$, where $\hat t(t,x)$ is Brownian local time, is determined uniquely by the Laplace transform $\int_0^\infty e^{-\lambda t}E\{e^{-\mu\hat t(t,r)-\eta\hat t(t,q)},\sup_{0\le s\le t}(\hat t(s,q)-\hat t(s,r))>h|w(0)=x\}\,dt.$ The computation of this transform constitutes the basic content of this paper. The obtained expression is used for the derivation of the exact modulus of continuity of the process $\hat t(t,x)$ with respect to the variable $x$:
$$ P\Big\{\limsup_{\substack{|y-x|=\Delta\downarrow0\\y,x\in R^1}}\frac{\sup_{0\le s\le t}|\hat t(s,y)-\hat t(s,x)|}{((\hat t(t,x)+\hat t(t,y))\Delta\ln 1/\Delta)^{1/2}}=2\Bigl\}=1. $$
English version:
Journal of Soviet Mathematics, 1987, Volume 36, Issue 4, Pages 439–451
DOI: https://doi.org/10.1007/BF01663452
Bibliographic databases:
Document Type: Article
UDC: 519.2
Language: Russian
Citation: A. N. Borodin, “Distribution of the supremum of increments of Brownian local time”, Problems of the theory of probability distributions. Part IX, Zap. Nauchn. Sem. LOMI, 142, "Nauka", Leningrad. Otdel., Leningrad, 1985, 6–24; J. Soviet Math., 36:4 (1987), 439–451
Citation in format AMSBIB
\Bibitem{Bor85}
\by A.~N.~Borodin
\paper Distribution of the supremum of increments of Brownian local time
\inbook Problems of the theory of probability distributions. Part~IX
\serial Zap. Nauchn. Sem. LOMI
\yr 1985
\vol 142
\pages 6--24
\publ "Nauka", Leningrad. Otdel.
\publaddr Leningrad
\mathnet{http://mi.mathnet.ru/znsl4224}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=788182}
\zmath{https://zbmath.org/?q=an:0569.60077}
\transl
\jour J. Soviet Math.
\yr 1987
\vol 36
\issue 4
\pages 439--451
\crossref{https://doi.org/10.1007/BF01663452}
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  • This publication is cited in the following 2 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Записки научных семинаров ПОМИ
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