Geodesic
Distribution of the supremum of increments of Brownian local time
Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part IX, Tome 142 (1985), pp. 6-24

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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 $x$: $$ P\Big\{\limsup_{\substack{|y-x|=\Delta\downarrow0\\y,x\in R^1}}\frac{\sup_{0\le s\le t}|\hat t(s,y)-\hat t(s,x)|}{((\hat t(t,x)+\hat t(t,y))\Delta\ln 1/\Delta)^{1/2}}=2\Bigl\}=1. $$ 
@article{ZNSL_1985_142_a0,
     author = {A. N. Borodin},
     title = {Distribution of the supremum of increments of {Brownian} local time},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {6--24},
     publisher = {mathdoc},
     volume = {142},
     year = {1985},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1985_142_a0/}
}
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A. N. Borodin. Distribution of the supremum of increments of Brownian local time. Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part IX, Tome 142 (1985), pp. 6-24. http://geodesic.mathdoc.fr/item/ZNSL_1985_142_a0/