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Explore a mathematical conference talk that establishes a primal-dual formulation for continuous-time mean field games and provides a complete analytical characterization of Nash equilibria. Learn how the representative player's control problem with measurable coefficients can be reformulated as a linear program over the space of occupation measures, and discover the dual formulation as a maximization problem over smooth subsolutions of the associated Hamilton-Jacobi-Bellman equation. Understand how this dual formulation plays a fundamental role in characterizing Nash equilibria of mean field games, and examine the complete characterization of all Nash equilibria through strong duality between the linear program and its dual problem. Gain insights into the solvability of the dual problem through analysis of the regularity of the associated HJB equation, with particular emphasis on how this characterization remains applicable even when the HJB equation lacks classical or continuous solutions and does not require convexity of the associated Hamiltonian or uniqueness of its optimizer. This 43-minute presentation was delivered at the Centre International de Rencontres Mathématiques during the thematic meeting "Probability, finance and signal: conference in honour of René Carmona" and offers advanced mathematical insights into mean field game theory and its applications.
