[For the entire collection see Zbl 0619.00019.]
Let \(P_ x\) be the probability measure associated with a standard Brownian motion B when started at x and let \(L_ t\) denote its local time at zero and \(\Gamma_ t\) the occupation time of \([0,+\infty)\). Put \(M_ t=\sup_{s\leq t}B_ s\), and define \(T=\inf \{s:\) \(B_ s=M_ t\}\). It has been noted in a paper by I. Karatzas and S. E. Shreve [Ann. Probab. 12, 819-828 (1984; Zbl 0544.60069)] that the joint \(P_ 0\)-distribution of \(L_ t\), \(\Gamma_ t\) and \(B_ t\) coincides in the case \(B_ t<0\) with that of \(M_ t\), T and \(B_ t.\)
In this paper the author offers a probabilistic explanation based on the excursion theory and path decompositions for this coincidence. The approach presented by the author is, roughly speaking, that a Brownian path is dissected, and re-assembled in such a way that local time becomes maximum and occupation time becomes location of maximum.