This work is about convergence to equilibrium problems for equations coming from kinetic theory. The bulk of the work is about Hypocoercivity. Hypocoercivity is the phenomenon when a semigroup shows exponentially relaxation towards equilibrium without the corresponding coercivity (dissipativity) inequality on the Dirichlet form in the natural space, i.e. a lack of contractivity. In this work we look at showing hypocoercivity in weak measure distances, and using probabilistic techniques. First we review the history of convergence to equilibrium for kinetic equations, particularly for spatially inhomogeneous kinetic theory (Boltzmann and Fokker-Planck equations) which motivates hypocoercivity. We also review the existing work on showing hypocoercivity using probabilistic techniques. We then present three different ways of showing hypocoercivity using stochastic tools. First we study the kinetic Fokker-Planck equation on the torus. We give two different coupling strategies to show convergence in Wasserstein distance, $W_2$. The first relies on explicitly solving the stochastic differential equation. In the second we couple the driving Brownian motions of two solutions with different initial data, in a well chosen way, to show convergence. Next we look at a classical tool to show convergence to equilibrium for Markov processes, Harris's theorem. We use this to show quantitative convergence to equilibrium for three Markov jump processes coming from kinetic theory: the linear relaxation/BGK equation, the linear Boltzmann equation, and a jump process which is similar to the kinetic Fokker-Planck equation. We show convergence to equilibrium for these equations in total variation or weighted total variation norms. Lastly, we revisit a version of Harris's theorem in Wasserstein distance due to Hairer and Mattingly and use this to show quantitative hypocoercivity for the kinetic Fokker-Planck equation with a confining potential via Malliavin calculus. We also look at showing hypocoercivity in relative entropy. In his seminal work work on hypocoercivity Villani obtained results on hypocoercivity in relative entropy for the kinetic Fokker-Planck equation. We review this and subsequent work on hypocoercivity in relative entropy which is restricted to diffusions. We show entropic hypocoercivity for the linear relaxation Boltzmann equation on the torus which is a non-local collision equation. Here we can work around issues arising from the fact that the equation is not in the H\"{o}rmander sum of squares form used by Villani, by carefully modulating the entropy with hydrodynamical quantities. We also briefly review the work of others to show a similar result for a close to quadratic confining potential and then show hypocoercivity for the linear Boltzmann equation with close to quadratic confining potential using similar techniques. We also look at convergence to equilibrium for Kac's model coupled to a non-equilibrium thermostat. Here the equation is directly coercive rather than hypocoercive. We show existence and uniqueness of a steady state for this model. We then show that the solution will converge exponentially fast towards this steady state both in the GTW metric (a weak measure distance based on Fourier transforms) and in $W_2$. We study how these metrics behave with the dimension of the state space in order to get rates of convergence for the first marginal which are uniform in the number of particles.

Neste trabalho consideramos um processo de Markov a tempo contínuo com espaço de estados finito em um meio aleatório, a saber, a dinâmica de Metropolis para o Modelo de Energia Aleatória Generalizado (GREM) com um número de níveis finito e discutimos o comportamento do seu tempo de chegada ao equilíbrio, o qual é dado pelo inverso da lacuna espectral de sua matriz de probabilidades de transição. No principal resultado desta tese provamos que o quociente entre o volume do sistema e o logaritmo do inverso da lacuna é quase sempre limitado, por cima, por uma função da temperatura, que também é a que descreve a energia livre do GREM sob o regime de temperaturas baixas. Como um estudo adicional, também é discutido um correspondente limitante inferior em um caso particular do GREM com 2 níveis. / In this work we consider a finite state continuous-time Markov process in a random environment, namely, the Metropolis dynamics for the Generalized Random Energy Model (GREM) with a finite number of levels, and we discuss the behavior of its reaching time to equilibrium which is given by inverse of the spectral gap of its transition probability matrix. On the main result of this thesis, we prove the division between the system volume and the logarithm of the inverse of the gap is almost surely upper bounded by a function of the temperature that it is also the function that describe the free energy of the GREM at low temperature. As an additional study, it is also discuss the corresponding limiting lower in a particular case of the 2-level GREM.

Neste trabalho consideramos um processo de Markov a tempo contínuo com espaço de estados finito em um meio aleatório, a saber, a dinâmica de Metropolis para o Modelo de Energia Aleatória Generalizado (GREM) com um número de níveis finito e discutimos o comportamento do seu tempo de chegada ao equilíbrio, o qual é dado pelo inverso da lacuna espectral de sua matriz de probabilidades de transição. No principal resultado desta tese provamos que o quociente entre o volume do sistema e o logaritmo do inverso da lacuna é quase sempre limitado, por cima, por uma função da temperatura, que também é a que descreve a energia livre do GREM sob o regime de temperaturas baixas. Como um estudo adicional, também é discutido um correspondente limitante inferior em um caso particular do GREM com 2 níveis. / In this work we consider a finite state continuous-time Markov process in a random environment, namely, the Metropolis dynamics for the Generalized Random Energy Model (GREM) with a finite number of levels, and we discuss the behavior of its reaching time to equilibrium which is given by inverse of the spectral gap of its transition probability matrix. On the main result of this thesis, we prove the division between the system volume and the logarithm of the inverse of the gap is almost surely upper bounded by a function of the temperature that it is also the function that describe the free energy of the GREM at low temperature. As an additional study, it is also discuss the corresponding limiting lower in a particular case of the 2-level GREM.