The distribution of the maximum of independent resetting Brownian motions

Abstract

The probability distribution of the maximum $M_t$ of a single resetting Brownian motion (RBM) of duration $t$ and resetting rate $r$, properly centred and scaled, is known to converge to the standard Gumbel distribution of the classical extreme value theory. This Gumbel law describes the typical fluctuations of $M_t$ around its average $\sim \ln (r t)$ for large $t$ on a scale of $O(1)$. Here we compute the large-deviation tails of this distribution when $M_t = O(t)$ and show that the large-deviation function has a singularity where the second derivative is discontinuous, signalling a dynamical phase transition. Then we consider a collection of independent RBMs with initial (and resetting) positions uniformly distributed with a density $ρ$ over the negative half-line. We show that the fluctuations in the initial positions of the particles modify the distribution of $M_t$. The average over the initial conditions can be performed in two different ways, in analogy with disordered systems: (i) the annealed case where one averages over all possible initial conditions and (ii) the quenched case where one considers only the contributions coming from typical initial configurations. We show that in the annealed case, the limiting distribution of the maximum is characterized by a new scaling function, different from the Gumbel law but the large-deviation function remains the same as in the single particle case. In contrast, for the quenched case, the limiting (typical) distribution remains Gumbel but the large-deviation behaviors are new and nontrivial. Our analytical results, both for the typical as well as for the large-deviation regime of $M_t$, are verified numerically with extremely high precision, down to $10^{-250}$ for the probability density of $M_t$.

Figures (12)

• Figure 1: Schematic trajectory of a stochastic search process $x(t)$, starting from $x(0)=0$, with global maximal value $M_t$ up to time $t$. A fixed target, shown by vertical dashed (red) line, is located at $M$. The survival probability $Q(M,t)$ of the target up to time $t$ is exactly the probability of the event $M_t \leq M$, as in Eq. (\ref{['max.2']}).
• Figure 2: A schematic representation of the two time scales associated with the fluctuations of the value of the maximum $M$ up to time $t$ for a resetting Brownian motion. For a fixed but large $t$, while the typical fluctuations scale as $\ln (rt)$ (shown by the red dashed curve), the large deviations scale linearly with $t$ as shown by the blue dashed line.
• Figure 3: A schematic plot of the PDF $P_r(M,t)$ of the maximum $M_t$ as a function of $M$, for fixed but large $t$ as in Eqs. \ref{['summary_PDF']} and \ref{['2_3']}. The two vertical dashed blue lines represent the typical and the large-deviation scales.
• Figure 4: Schematic trajectories of $4$ independent resetting Brownian motions up to time $t$, each starting and resetting at $x_i \leq 0$ ($i=1$, $2$, $3$ and $4$). The target is located at $M>0$ shown by the dashed red vertical line.
• Figure 5: A schematic plot of the annealed PDF $P_{\rm an}(M,t)$ vs. $M$ in Eq. \ref{['PDF_an_summary']}. The two vertical dashed blue lines represent the typical and the large-deviation scales.