Applied Markov Chain Tree Theorem

This doctoral thesis explores the multifaceted applications of the Markov chain tree Theorem in the study of stochastic processes, highlighting the deep connection between graph theory and the dynamics of continuous-time Markov chains. The work is organized into two main parts, each addressing asymptotic problems through a combinatorial lens. In the First Part, we analyze the behavior of Markov chains on finite directed graphs where transition rates decay exponentially with respect to a positive parameter. We introduce the discrete counterpart of the Freidlin-Wentzell theory, formulating a discrete Hamilton-Jacobi equation for the large deviations functional of the invariant measure. Through a geometric characterization of viscosity solutions and the application of the Matrix tree Theorem, we classify all the solutions and establish a selection principle analogous to that observed in continuous diffusion processes. The Second Part is devoted to the study of multiscale Markov chains and metastability phenomena. Leveraging the theory of trace processes, the thesis describes the decomposition of the limiting invariant measure through effective dynamics on reduced state spaces. By extending the Markov chain tree Theorem to chains that are not necessarily irreducible, the entire problem of metastability is framed within a purely combinatorial context based on the relationships between the weights of arborescences and directed forests. Overall, the thesis demonstrates how the topological structure of the transition graph provides powerful and rigorous tools for resolving complex issues related to convergence and scale separation in stochastic systems.