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Zapiski Nauchnykh Seminarov LOMI, 1985, Volume 142, Pages 6–24
(Mi znsl4224)
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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
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 $x$:
$$
P\Big\{\limsup_{\substack{|y-x|=\Delta\downarrow0\\y,x\in R^1}}\frac{\sup_{0\le s\le t}|\hat t(s,y)-\hat t(s,x)|}{((\hat t(t,x)+\hat t(t,y))\Delta\ln 1/\Delta)^{1/2}}=2\Bigl\}=1.
$$
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
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https://www.mathnet.ru/eng/znsl4224 https://www.mathnet.ru/eng/znsl/v142/p6
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