From PDEs to Probability: Hamilton-Jacobi equations and Large Deviations

Abstract

This thesis focuses on two main topics: large deviations for Markov processes and the well-posedness of Hamilton–Jacobi equations. The first two chapters provide an introduction to both areas. Chapter 1 explores the mathematical foundations of Hamilton–Jacobi equations, highlighting their applications in control theory and emphasising the role of viscosity solutions in handling situations where classical solutions fail. Chapter 2 introduces large deviations theory, starting from basic examples and leading to rigorous definitions. A key theme is the connection between large deviations and Hamilton–Jacobi equations, introduced through the Feng–Kurtz method. The subsequent chapters present the main research contributions of this thesis. Chapter 3 studies two examples of two-scale Markov processes and applies the Feng–Kurtz method to establish a large deviations principle. Chapter 4 transitions to the second theme of this thesis: the well-posedness of Hamil- ton–Jacobi equations. Motivated by the previous examples, we analyse a general class of Hamilton–Jacobi equations. We establish a comparison principle for viscosity solutions, demonstrating its applicability in a broad setting. Chapter 5 extends these results by proving the existence of viscosity solutions for a general class of Hamilton–Jacobi equations using Lyapunov control techniques. In the final chapter, Chapter 6, we investigate second-order Hamilton–Jacobi equations, presenting a novel proof of the comparison principle for viscosity solutions.