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This thesis develops the theoretic mathematical foundations of Fokker-Planck-Kolmogorov operators within a potential theory framework to study metastability in stochastic processes. These operators generate Langevin dynamics, which model the motion of a particle influenced by thermal noise and an external potential field. A key focus of this study is the metastable transition time, the time required for a particle to move between local minima, which plays a fundamental role in applications such as chemical reactions, quantum tunneling, and neural network training.

The main contributions of this thesis include a comprehensive analysis of weak solutions to (possibly degenerate) Fokker-Planck-Kolmogorov equations. Specifically, we develop a Galerkin method for solving the Cauchy problem in a periodic setting, establish the existence and uniqueness of weak solutions for the stationary Kolmogorov operator in bounded product domains, and introduce Perron's solutions in general bounded domains. Additionally, we prove global boundedness results for time-dependent solutions in bounded time cylinders. Finally, we develop a variational framework for potential theory for stationary Kolmogorov operators. These results provide new mathematical tools for studying metastability and deepen our understanding of stochastic systems while also opening new research directions in Kolmogorov equations, including higher boundary regularity and obstacle problems.