Lecture Notes on Gradient Flows and Optimal Transport

Abstract

We present a short overview on the strongest variational formulation for gradient flows of geodesically $\lambda$-convex functionals in metric spaces, with applications to diffusion equations in Wasserstein spaces of probability measures. These notes are based on a series of lectures given by the second author for the Summer School "Optimal transportation: Theory and applications" in Grenoble during the week of June 22-26, 2009.

References in corpus (3)

• Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory
• A Wasserstein approach to the one-dimensional sticky particle system
• Contraction of general transportation costs along solutions to Fokker-Planck equations with monotone drifts

Cited by in corpus (8)

• Gradient flow structure for McKean-Vlasov equations on discrete spaces
• Barriers of the McKean--Vlasov energy via a mountain pass theorem in the space of probability measures
• Functional Gradient Boosting based on Residual Network Perception
• A proof of the mean-field limit for $\lambda$-convex potentials by $\Gamma$-Convergence
• An invariance principle for gradient flows in the space of probability measures
• Structural Compactness and Stability of Pseudo-Monotone Flows
• Geometrically-exact time-integration mesh-free schemes for advection-diffusion problems derived from optimal transportation theory and their connection with particle methods
• Structural stability of flows via evolutionary Gamma-convergence of weak-type