Convex functionals of probability measures and nonlinear diffusions on manifolds

By Karl‐Theodor Sturm. 2005

115 citations recorded by the source corpus.

Subfields: Applied Mathematics, Cardiology and Cardiovascular Medicine, Cell Biology

View Paper: https://www.semanticscholar.org/paper/9854054c3ad7f9afc7036a86c28c7e7939b64361

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Cardiology and Cardiovascular Medicine

In Cell Biology

Citations in the corpus

On the geometry of metric measure spaces. II (2006, background): “Independently of [ St05 ] and [St04], the particular cases CD(K;1) and CD(0;N) of the curvaturedimension condition | deflned in essentially the same form | were also discussed in a recent paper [LV04] by John Lott and C¶edric Villani.”
Prekopa-Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport (2006, result): “We recently learned that the next result was independently proved by Sturm [40] and by Lott and Villani [25] along their investigation of curvature in abstract length spaces.”
Ricci curvature for metric-measure spaces via optimal transport (2004, methodology): “After the paper was finished we learned of related work by Karl-Theodor Sturm [38, 39].”
Characterization of absolutely continuous curves in Wasserstein spaces (2006, background): “Indeed, many papers have recently appeared dealing with various aspects of measure metric spaces or, in particular, of Riemannian manifolds, strictly connected to Kantorovitch–Rubinstein–Wasserstein distance (see [21,22,30,31], in metric measure spaces, and [14,25,26,29,33]).”
Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces (2009, background): “By the result of [30], defining VG(x) := 12 log(detG(x)), the functional φ(μ) := ∫”
Sharp logarithmic Sobolev inequalities on gradient solitons and applications (2008, background): “Recently, several authors [24, 35] based on early works [14, 25], see [38, Chapter 17] for a whole account of the history, have characterized curvaturedimension bounds in terms of the displacement convexity of the Boltzmann relative entropy functional.”
Contractions in the 2-Wasserstein Length Space and Thermalization of Granular Media (2006, result): “The previous result holds for nonlinearities A(ρ) satisfying (49) and the additional hypotheses in [22, Theorem 18], for instance, P( ρ)satisfying P � (0) = 0, P ∈ C 3 (0, ∞) and P �� 0. Related contractivity results for the heat equation [47], porous medium equation [ 44 ], and general gradient flo”
A Family of Nonlinear Fourth Order Equations of Gradient Flow Type (2009, background): “[37] Sturm, K.-T. Convex functionals of probability measures and nonlinear diffusions on manifolds.”
Displacement convexity of entropy and related inequalities on graphs (2012, background): “A remarkable and powerful fact is that, when X is a Riemannian manifold, one can relate the Ricci curvature of the space to the convexity of entropy along geod esics [34, 8, 43, 29, 51, 54].”