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We establish the boundedness of weak subsolutions for a class of degenerate Kolmogorov equations of the hypoelliptic type, compatible with a homogeneous Lie group structure, within bounded product domains using the De Giorgi iteration. We employ the renormalization formula to handle boundary values and provide energy estimates. An – type embedding estimate derived from the fundamental solution is utilized to incorporate lower-order divergence terms. This work naturally extends the boundedness theory for uniformly parabolic equations, with matching exponents for the coefficients.

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.

n this paper, we investigate weak solutions and Perron-Wiener-Brelot solutions to the linear stationary Kramers-Fokker-Planck equation in bounded domains. We establish the existence of weak solutions in product domains by applying the Lions-Lax-Milgram theorem and the vanishing viscosity method. Furthermore, we show that these solutions coincide in well-behaved domains. Building on the existence of weak solutions in product domains, we develop the foundational theory of Perron-Wiener-Brelot solutions in arbitrary bounded domains. Our results rely on recent advancements in the theory of kinetic Fokker-Planck equations with rough coefficients.

In this paper, we develop a Galerkin-type approximation, with quantitative error estimates, for weak solutions to the Cauchy problem for kinetic Fokker-Planck equations in the domain (0,T)×D×Rd, where D is either Td or Rd. Our approach is based on a Hermite expansion in the velocity variable only, with a hyperbolic system that appears as the truncation of the Brinkman hierarchy, as well as ideas from $\href{arXiv:1902.04037v2}{Alb+21}$ and additional energy-type estimates that we have developed. We also establish the regularity of the solution based on the regularity of the initial data and the source term.

We introduce a variational formulation for a general class of possibly degenerate, non-self-adjoint Fokker-Planck operators in divergence form, motivated by the work of Albritton et al. (2024), and prove that it is suitable for defining the variational capacity.  Using this framework, we establish rough estimates for the equilibrium potential in the elliptic case, providing a novel approach compared to previous methods.  Finally, we derive the Eyring-Kramers formula for non-self-adjoint elliptic Fokker-Planck operators in divergence form, extending the results of Landim et al. (2019) and Lee & Seo (2022).