Brownian behaviour of the Riemann zeta function around the critical line

We establish a Brownian extension to Selberg’s central limit theorem for the Riemann zeta function. This implies various limiting distributions for $\zeta$, including an analogue of the reflection principle for the maximum of the Brownian motion: as $T$ diverges, for any $u>0$ we have $$\frac{1}{T}\cdot \mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}\{0\le t\le T:\underset{\sigma \ge \frac{1}{2}}{\max }\log \mathrm{\mid }\zeta (\sigma +it)\mathrm{\mid }\ge u\sqrt{\frac{1}{2}\log \log T}\}\to 2{\displaystyle {\int }_u^{\mathrm{\infty }}\frac{e^{-\frac{x^2}{2}}}{\sqrt{2\pi }}\mathrm{d}x\mathrm{.}}$$