An integral test for the supremum of Wiener local time (Q1123485)

Let L(x,t) (x\(\in {\mathbb{R}}\), \(t\geq 0)\), be the jointly continuous Brownian local time, and introduce \(L^*(t)=\sup_{x\in {\mathbb{R}}}L(x,t)\). A positive nondecreasing function a belongs to the upper-upper class (respectively the upper-lower class) of \(L^*\) if \(L^*(t)\leq a(t)\) for all large enough t a.s. (respectively if for a.e. \(\omega\), there is a sequence \(t_ i=t_ i(\omega)\), lim \(t_ i=\infty\), such that, for every integer i, \(L^*(t_ i)>a(t))\). These classes are characterized by an integral test.