formula

$Ee^{\alpha X_t}&=&\int_{\theta =0}^{2\pi}\int_{r=0}^{\infty}r^{\beta +1}\frac{1}{2\pi t}\exp\bigg(-\frac{r^2}{2t}\bigg)drd\theta\\ &=&\frac{1}{2t}\int_0^{\infty}s^{(\beta/2+1)-1}\exp \bigg(-\frac{s}{2t}\bigg)dr\quad (s=r^2)\\ &=&\frac{\Gamma (\beta /2+1)}{(2t)^{\beta /2+2}}<\infty$