engleski

Zero-One law for some Brownian Functionals

We prove that for $a>0$, $(B_t)$ one-dimensional Brownian motion and ${\tau}_0=\inf \{t>0: B_t=0\}$ the following zero-one law is valid $$ P^a[\int_0^{{\tau}_0} \frac{ds}{B_s^{\alpha}} < +\infty]= 1, if 0<\alpha <2; 0 if alpha \geq 2 $$

Brownian motion; Brownian functional; zero-one law

nije evidentirano