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We consider a Brownian motion (BM) $x(\tau)$ and its maximal value $x_{\max}= \max_{0 \leq \tau \leq t} x(\tau)$ on a fixed time interval $[0,t]$. We studyfunctionals of the maximum of the BM, of the form ${\calO}_{\max}(t)=\int_0^t\, V(x_{\max} - x(\tau)) {\rm d} \tau$ where $V(x)$ can beany arbitrary function and develop various analytical tools to compute theirstatistical properties. These tools rely in particular on (i) a "countingpaths" method and (ii) a path-integral approach. In particular, we focus on thecase where $V(x) = \delta(x-r)$, with $r$ a real parameter, which is relevantto study the density of near-extreme values of the BM (the so called density ofstates), $\rho(r,t)$, which is the local time of the BM spent at given distance$r$ from the maximum. We also provide a thorough analysis of the family offunctionals ${T}_{\alpha}(t)=\int_0^t (x_{\max} - x(\tau))^\alpha \, {\rmd}\tau$, corresponding to $V(x) = x^\alpha$, with $\alpha$ real. As $\alpha$ isvaried, $T_\alpha(t)$ interpolates between different interesting observables.For instance, for $\alpha =1$, $T_{\alpha = 1}(t)$ is a random variable of the"area", or "Airy", type while for $\alpha=-1/2$ it corresponds to the maximumtime spent by a ballistic particle through a Brownian random potential. On theother hand, for $\alpha = -1$, it corresponds to the cost of the optimalalgorithm to find the maximum of a discrete random walk, proposed by Odlyzko.We revisit here, using tools of theoretical physics, the statistical propertiesof this algorithm which had been studied before using probabilistic methods.Finally, we extend our methods to constrained BM, including in particular theBrownian bridge, i.e., the Brownian motion starting and ending at the origin.

On certain functionals of the maximum of Brownian motion and their applications