Large Deviations for Markov Jump Processes and Hamiltonian Trajectories

Abstract

This master thesis is concerned with Large Deviation Theory in combination with Lagrangian and Hamiltonian dynamics. In particular, the Large Deviation behaviour of the empirical distribution of n independent two-state continuous-time Markov processes is studied. We start by looking at the general theory of Large Deviations in both the finite dimensional case as well as for infinite dimensional stochastic processes. After this, the connection is shown of Large Deviation Theory with Lagrangian and Hamiltonian dynamics. The Hamiltonian of the empirical distribution of n independent two-state Markov processes is derived and using this, via the Hamilton equations, the dynamics of this process are derived. That is, the most likely path that is taken when in time T we force the path to start in state a and end in state b. The main goal of this thesis is to find out more about the dynamics of this process (behaviour of the trajectories) and to derive an explicit equation for the so-called Action integral. We want to compute the Action for the general case. That is, the case in which the rate going from state one to state two can differ from the rate going from state two to state one. Once the Action integral is computed we look at the asymptotic behaviour of this Action integral. This is important as it says something about the probability of some trajectories occurring for T an extreme. We look at the asymptotics for both T in the limit to zero and T in the limit to infinity.