Brownian local times and taboo processes

Introduction.This paper is an application of the Markov property of the Brownian local times, as obtained in ([4], [6]; see also [10]), to the study of the sojourn time distributions of the one-dimensional Brownian motion (Wiener process).We recall the following two facts.If X(t), Z(0)=0, denotes the Wiener process (with continuous path functions), /^-ijY..^* is the "local time" (which by [9] exists and is continuous in (t, x) except for an exceptional set of paths having probability 0), and if for a>0 we set Fa(a) = inf {t : f(t, a, w)>a} then f(T0(a), x, w), x^O, is the diffusion process with x as "time parameter," infinitesimal generator of the form 4y(d2)2, initial value a, and absorbing barrier at 0 [4, p. 1].Furthermore, for a > 0, f(Ta(a), a -x, w), 0xa, is the restriction to parameter values in [0, a] of the diffusion process with generator 4y(d22) + 2(d) and initial value a, where 0 is an entrance bound-