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

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 study functionals of the maximum of the BM, of the form ${\cal O}_{\max}(t)=\int_0^t\, V(x_{\max} - x(\tau)) {\rm d} \tau$ where $V(x)$ can be any arbitrary function and develop various analytical tools to compute their statistical properties. These tools rely in particular on (i) a "counting paths" method and (ii) a path-integral approach. In particular, we focus on the case where $V(x) = \delta(x-r)$, with $r$ a real parameter, which is relevant to study the density of near-extreme values of the BM (the so called density of states), $\rho(r,t)$, 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 of functionals ${T}_{\alpha}(t)=\int_0^t (x_{\max} - x(\tau))^\alpha \, {\rm d}\tau$, corresponding to $V(x) = x^\alpha$, with $\alpha$ real. As $\alpha$ is varied, $T_\alpha(t)$ 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 maximum time spent by a ballistic particle through a Brownian random potential. On the other hand, for $\alpha = -1$, it corresponds to the cost of the optimal algorithm to find the maximum of a discrete random walk, proposed by Odlyzko. We revisit here, using tools of theoretical physics, the statistical properties of this algorithm which had been studied before using probabilistic methods. Finally, we extend our methods to constrained BM, including in particular the Brownian bridge, i.e., the Brownian motion starting and ending at the origin.