A decomposition of the Brownian path
The Brownian path \(\{\omega\) (s),s\(\leq t\}\) is transformed into the path \(\{\) \({\hat \omega}\)(s),s\(\leq t\}\) such that the resulting process is again Brownian on [0,t] and \[ ({\hat \omega}(t),\gamma_ t({\hat \omega}),M(t,{\hat \omega},\sigma_ t({\hat \omega}))=(\omega (t),\gamma_ t(\omega),L(t,\omega),\Gamma_+(t,\omega)) \] on \(\{\omega (t)<0\}\), where L(t,\(\omega)\) is local time of the origin, \(\gamma_ t(\omega)=\sup \{s\leq t:\omega (s)=0\}\), \(\Gamma_+(t,\omega)=meas\{s\in [0,t]:\omega\) (s)\(\geq 0\}\), M(t,\({\hat \omega}\))\(=\max \{{\hat \omega}(s)\); \(s\leq t\},\) \(\sigma_ t({\hat \omega})\) the (a.s. unique) location where M(t,\({\hat \omega}\)) is achieved. Several consequences are considered.
scientific article; zbMATH DE number 3996802