UCLA

This thesis explores several problems in the calculus of variations especially in the space of measures. Drawing from mean field games (MFG), Hamilton--Jacobi equations (HJE), optimal transport (OT), and quantum mechanics, we explore challenges in existence, uniqueness, and well-posedness in both deterministic and stochastic settings.

The first part of this work examines mean field games and the associated master equation, an infinite-dimensional partial differential equation that links individual decisions with population dynamics. We establish new well-posedness results under monotonicity and convexity conditions, extending previous theories to accommodate broader interaction structures and types of noise (in particular we obtain results in the absence of idiosyncratic noise).

Next, we investigate Hamilton--Jacobi equations in optimal control and classical mechanics, using canonical transformations as a method to achieve global well-posedness in non-convex settings. This approach expands the range of systems that can be studied, with implications for stability and optimal trajectory analysis. This method also extends to the master equation, revealing hidden monotonicity properties that result in new well-posedness theories.

Finally, we explore an extension OT to quantum mechanics by formulating a quantum dynamic transport problem governed by the Schrödinger equation. By establishing a link with the Pauli problem in quantum state reconstruction, this framework could open new avenues for attacking this long-standing problem.