A decomposition of the Brownian path

Abstract

The Brownian path {[omega](s); 0 [less-than-or-equals, slant] s [less-than-or-equals, slant] t} is dissected and then reassembled in such a way that (i) the last visit [gamma]t at the origin, as well as the fragment {[omega](s); [gamma]t [less-than-or-equals, slant] s [less-than-or-equals, slant] t}, are left invariant; (ii) on [0, [gamma]t], local time becomes maximum-to-date and occupation time ofR+ becomes location of maximum; and (iii) the resulting process is again Brownian. Characterizations of conditional processes are employed to establish the result. Several consequences of the latter are discussed.60J65 60G17 Brownian motion and bridge local time occupation time conditioning path decomposition