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 \[ \frac{1}{T}\cdot {\rm meas}\Big\{0\leq t\leq T:\max_{\sigma\geq \tfrac{1}{2}}\log|\zeta(\sigma+i t)|\geq u \sqrt{\tfrac{1}{2}\log \log T} \Big\}\to 2 \displaystyle\int_u^{\infty} \frac{e^{-\frac{x^2}{2}}}{\sqrt{2\pi}}\mathrm{d} x. \]