Rare events in one dimension: transport and first passage.
Ivan Burenev
This thesis studies the statistics of rare events in one-dimensional stochastic systems. We focus on two physical systems: a system of non-interacting Brownian particles on the line with a steplike initial condition, and a jump-drift process. For each, we combine analytical and numerical methods to obtain precise quantitative results that go far beyond what Gaussian approximations can provide. In addition, we develop an importance sampling strategy, the local tilt, for the numerical study of counting statistics in one-dimensional systems.
For the Brownian particle system, we focus on two observables: the local time density at the origin and the occupation time on the positive half line. A peculiar feature of such systems is that the statistics retain a long-time memory of the initialization, which persists even at arbitrary large observation times. We compute the means and variances of both observables and derive the corresponding large deviation rate functions for the quenched and annealed averaging schemes.
For the jump-drift process, we study the first-passage properties. We introduce a mapping onto an effective discrete-time random walk and obtain the joint distribution of the first-passage time and the number of jumps. We establish a phase diagram with two regimes separated by a critical point: the survival (weak drift) regime where the process has a finite probability of never crossing the origin, and the absorption (strong drift) regime where first passage surely occurs. We confirm that this behavior holds for arbitrary light-tailed distributions.
Jury : Roberto Artuso, Olivier Bénichou, Denis Grebenkov (rapporteur), Michael Kearney (invité), Satya Majumdar (directeur de thèse), Cécile Monthus, Clément Sire (rapporteur)