Distribution of the supremum of increments of Brownian local time

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 $x$:
$$ P\Big\{\limsup_{\substack{|y-x|=\Delta\downarrow0\\y,x\in R^1}}\frac{\sup_{0\le s\le t}|\hat t(s,y)-\hat t(s,x)|}{((\hat t(t,x)+\hat t(t,y))\Delta\ln 1/\Delta)^{1/2}}=2\Bigl\}=1. $$