Wasserstein Gradient Flows and Otto Calculus

Felix Otto’s pioneering work introduced a geometric perspective on the porous medium equation, interpreting it as a gradient flow in the space of absolutely continuous probability measures equipped with the Wasserstein metric. This insight led to the rigorous framework developed by Ambrosio, Gigli, and Savaré, which formalizes Wasserstein gradient flows and extends Otto’s asymptotic estimates to a broader class of dissipative equations. This thesis explores key aspects of this theory, starting with gradient flows in Hilbert spaces before extending these ideas to the Wasserstein space. We discuss the continuity equation, the Benamou-Brenier formula, and the characterization of tangent vectors in the Wasserstein space. The study then turns to displacement convexity and the subdifferential calculus necessary for defining Wasserstein gradient flows.