formula

$EX_1 &=& EX_t\quad (\forall{t}\geq 1)\\ &=& \int_{\theta =0}^{\theta = 2\pi}\int_{r=0}^{r=\infty}(-r\log r)\frac{1}{2\pi t}\exp \bigg( -\frac{r^2}{2t}\bigg)drd\theta\\ &\leq &\frac{1}{te}\int_{0}^{\infty}\exp\bigg(-\frac{r^2}{2t}\bigg)dr\\ &=&\frac{\sqrt{2\pi}}{2e\sqrt{t}}$