Characterization of absolutely continuous curves in Wasserstein spaces

By S. Lisini. 2006

137 citations recorded by the source corpus.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/dd647d5f6186142a2247bf00e3bf70f76d7427bd

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In Applied Mathematics

In Astronomy and Astrophysics

In Geometry and Topology

Citations in the corpus

Equivalent definitions of BV space and of total variation on metric measure spaces (2014, methodology): “(Approximating plans ΣN) We can argue as in [26], with minor changes, to build approximating plans ηN in this way: • we consider ΣN , for i = 0, .”
Fairness in Machine Learning (2020, background): “As R is a geodesic space, i.e. there exists a geodesic between every pair of points in that space, then so is P 2 ( R ) [110].”
Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling (2015, methodology): “The proof that we provide here is different from that of [17] and is based on the following represenatation Theorem (in the spirit of [213]).”
The Splitting Theorem in Non-Smooth Context (2013, unknown): “More generally, Lisini ([48]) proved that every absolutely continuos curve (µt) admits a lifting in the following sense: Theorem 2.2 Let (X, d) be a complete, separable metric space and (µt) ⊂ P2(X) a W2absolutely continuous curve such that ∫ 1 0 |µ̇t|2 dt ∞.”
Nonlinear mobility continuity equations and generalized displacement convexity (2009, methodology): “This dynamical formulation of the Wasserstein distance was rigorously established by Benamou and Brenier in [5] and extended to more general situations in [2] and [30].”
A User’s Guide to Optimal Transport (2013, unknown): “Lisini [54].”
Sobolev spaces in metric measure spaces: reflexivity and lower semicontinuity of slope (2012, background): “We will need the following result, proved in [27]: it shows how to associate to an absolutely continuous curve μt w.”
Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces (2009, background): “On the other hand, since RG is a geodesic space, by Proposition 1 in [24], P2(R n G) is a geodesic space too.”
The Wasserstein space of stochastic processes (2021, methodology): “As ZN = XN is a geodesic space, it follows from [56] that Pp(XN ) is a geodesic space, too.”
Metric measure spaces with Riemannian Ricci curvature bounded from below (2011, unknown): “ontinuous curve γ: [0,1] → X. Lemma 2.8 Let (X,d,m) ∈ Xand ψ: X→ RLipschitz. For all µt ∈ AC2([0,1];P 2(X)) it holds 1 Z X ψdµ − Z X ψdµ 0 1 ≤ Z 0 Z X |∇ψ|2dµ t 1/2 |µ˙ |dt. (2.15) Proof. Applying [25], we can ﬁnd a probability measure π in C([0,1];X) concentrated on AC2([0,1];X) = 0 and satisfying”