Stochastic processes and disordered systems, around Brownian motion
Mathieu Delorme (LPT)

In this thesis, we study stochastic processes appearing in different areas
of statistical physics: Firstly, fractional Brownian motion is a
generalization of the well-known Brownian motion to include memory. Memory
effects appear for example in complex systems and anomalous diffusion, and
are difficult to treat analytically, due to the absence of the Markov
property. We develop a perturbative expansion around standard Brownian
motion to obtain new results for this case. We focus on observables
related to extreme-value statistics, with links to mathematical objects:
Levy’s arcsine laws and Pickands’ constant.

Secondly, the model of elastic interfaces in disordered media is
investigated. We consider the case of a Brownian random disorder force. We
study avalanches, i.e. the response of the system to a kick, for which
several distributions of observables are calculated analytically. To do
so, the initial stochastic equation is solved using a deterministic
non-linear instanton equation. Avalanche observables are characterized by
power-law distributions at small-scale with universal exponents, for which
we give new results.