Tsobgni Nyawo Pelerine

Thesis Abstract We study in this thesis the fluctuations of time-integrated functionals of Markov processes, which represent physical observables that can be measured in time for noisy systems driven in nonequilibrium steady states. The goal of the thesis is to illustrate how techniques from the theory of large deviations can be used to obtain the probability distribution of these observables in the long -time limit through the knowledge of an important function, called the rate function. We also illustrate a recent theory of driven processes that aims to describe how fluctuations of observables are created in time by means of an effective process with modified forces or potentials. This is done by studying two simple models of nonequilibrium processes based on the Langevin equation. The first is a periodic diffusion that has current fluctuations, whereas the second is the simple drifted Brownian for which we study the occupation fluctuations. For these two models, we calculate analytically and numerically the rate function, as well as the associated driven process. The results for the periodic diffusion show, on the one hand, that there is Gaussian to non Gaussian crossover in the current fluctuations, which can be interpred from the driven process. On the other hand, the Brownian model provides the simplest examples of a dynamical phase transition, that is , a phase transition in the fluctuations of observables. Other connections with fluctuation retation, Josephson junctions and the geometric Brownian motion are discussed.