Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below

By Luigi Ambrosio, Nicola Gigli, Giuseppe Savar'e. 2011

564 citations recorded by the source corpus.

This paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces $(X,\mathsf {d},\mathfrak {m})$. Our main results are: A general study of the relations between the Hopf–Lax semigroup and Hamilton–Jacobi equation in metric spaces (X,d).The equivalence of the heat flow in $L^{2}(X,\mathfrak {m})$ generated by a suitable Dirichlet energy and the Wasserstein gradient flow of the relative entropy functional $\mathrm {Ent}_{\mathfrak {m}}$ in the space of probability measures .The proof of density in energy of Lipschitz functions in the Sobolev space $W^{1,2}(X,\mathsf {d},\mathfrak {m})$.A fine and very general analysis of the differentiability properties of a large class of Kantorovich potentials, in connection with the optimal transport problem, is the fourth achievement of the paper. Our results apply in particular to spaces satisfying Ricci curvature bounds in the sense of Lott and Villani (Ann. Math. 169:903–991, 2009) and Sturm (Acta Math. 196: 65–131, 2006, and Acta Math. 196:133–177, 2006) and require neither the doubling property nor the validity of the local Poincaré inequality.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b20c987e227a71e96fc08349b38f8e2d89fba883

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Citations in the corpus

From volume cone to metric cone in the nonsmooth setting (2015, methodology): “see [1] and [15] for more details and for the proof of the following equivalent characterization:”
Short-time behavior of the heat kernel and Weyl's law on $RCD^*(K, N)$-spaces (2017, background): “Let us recall basic facts about Sobolev spaces and heat flow in metric measure spaces (X,d,m), see [2] and [25] for a more systematic treatment of this topic.”
Ricci Curvature of Finite Markov Chains via Convexity of the Entropy (2011, background): “The theory of metric measure spaces with Ricci curvature bounds in the sense of Lott, Sturm, and Villani is still under active development [2, 3].”
An Analog of the 2-Wasserstein Metric in Non-Commutative Probability Under Which the Fermionic Fokker–Planck Equation is Gradient Flow for the Entropy (2012, background): “A similar gradient flow structure in the space of probability measures has meanwhile been developed and exploited in many different settings [1, 2, 6, 9, 10, 12, 14, 15, 20, 22, 23, 25, 26].”
Gradient flows of the entropy for jump processes (2012, background): “By now, similar interpretations of the heat flow have been established in a variety of settings ranging from Riemannian manifolds to abstract metric measure spaces, see [2,15,17,19,25].”