On pathwise projective invariance of Brownian motion. I (Q1110906)

Let \(\{B(t); t\in {\mathbb R}\}\) be a suitable version of the standard Brownian motion and \(g=\left( \begin{matrix} a,b\\ c,d\end{matrix} \right)\) be an element of \(\text{SL}(2,\mathbb R)\). Define an action of \(g\) on the path space, \[ B^ g(t;\omega)\equiv (ct+d)B((at+b)/(ct+d);\omega)-ctB(a/c;\omega)- dB(b/d;\omega). \] Then, \(B^ g\) is again a Brownian motion and the action above is compatible to the group action of \(\text{SL}(2,\mathbb R)\): \((B^ g)^ h(t;\omega)=B^{gh}(t;\omega).\) Fix an interval \([\alpha,\beta]\) and consider Lévy's normalized Brownian bridge: \[ \xi^{g,[\alpha,\beta]}(t)\equiv {\mathcal N}\{B^ g(t)-B^ g(\alpha)- {\mathbb E}[B^ g(t)-B^ g(\alpha)| B^ g(\beta)-B^ g(\alpha)]\}. \] Then the following pathwise relation holds \[ \xi^{g,[g\alpha,g\beta]}(gt;\omega)=\xi^{e,[\alpha,\beta]}(t;\omega). \] The above relation instructs the mysterious projective invariance of Lévy [see \textit{P. Lévy}, Processus stochastiques et mouvement brownien, Chap. 1-2 (1965; Zbl 0137.11602)].