Gradient flows and a Trotter-Kato formula of semi-convex functions on CAT(1)-spaces

Ohta, Shin‐ichi; Pálfia, Miklós

doi:10.1353/ajm.2017.0025

Cited by 21 publications

(14 citation statements)

References 30 publications

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“…The CAT(κ) case has been treated in [6], at least under some compactness assumptions on the sublevels of the functional. Such compactness assumption has been removed in [7].…”

Section: Proofmentioning

confidence: 99%

“…In this setting gradient flow trajectories (x t ) of E (or curves of maximal slopes) are defined by imposing the maximal rate of dissipation (3.1)]. It has been later understood [3][4][5][6][7] that if E is λ-convex and the metric space has some form of some Hilbert-like structure at small scales, then an equivalent formulation can be given via the so-called Evolution Variational Inequality d dt…”

Section: Introductionmentioning

confidence: 99%

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A Differential Perspective on Gradient Flows on $$\textsf {CAT} (\kappa )$$-Spaces and Applications

Gigli

Nobili

2021

J Geom Anal

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We review the theory of Gradient Flows in the framework of convex and lower semicontinuous functionals on $$\textsf {CAT} (\kappa )$$ CAT ( κ ) -spaces and prove that they can be characterized by the same differential inclusion $$y_t'\in -\partial ^-\textsf {E} (y_t)$$ y t ′ ∈ - ∂ - E ( y t ) one uses in the smooth setting and more precisely that $$y_t'$$ y t ′ selects the element of minimal norm in $$-\partial ^-\textsf {E} (y_t)$$ - ∂ - E ( y t ) . This generalizes previous results in this direction where the energy was also assumed to be Lipschitz. We then apply such result to the Korevaar–Schoen energy functional on the space of $$L^2$$ L 2 and (0) valued maps: we define the Laplacian of such $$L^2$$ L 2 map as the element of minimal norm in $$-\partial ^-\textsf {E} (u)$$ - ∂ - E ( u ) , provided it is not empty. The theory of gradient flows ensures that the set of maps admitting a Laplacian is $$L^2$$ L 2 -dense. Basic properties of this Laplacian are then studied.

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“…The CAT(κ) case has been treated in [6], at least under some compactness assumptions on the sublevels of the functional. Such compactness assumption has been removed in [7].…”

Section: Proofmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Differential Perspective on Gradient Flows on $$\textsf {CAT} (\kappa )$$-Spaces and Applications

Gigli

Nobili

2021

J Geom Anal

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“…is the gradient curve of the function f starting at x. As a reference one can use [OP17] or [Lyt05], see also [Pet07], [May98] and [AGS05] for a general theory of gradient flows in metric spaces.…”

Section: Preliminariesmentioning

confidence: 99%

Improvements of upper curvature bounds

Lytchak¹,

Stadler²

2020

Trans. Amer. Math. Soc.

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“…These metric approaches treat the sequence of inductive means S n as a discrete-time approximation of the gradient flow of the cost function to be minimized in (1), and apply the Riemannian-like nature of the CAT (1) property in an essential way. For further results on the continuous time metric theory of gradient flows see [1,3,29].…”

Section: The Contractive Barycenter Of Positive Operatorsmentioning

confidence: 99%

Strong law of large numbers for the $L^1$-Karcher mean

Lim¹,

Pálfia²

2019

Preprint

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Sturm's strong law of large numbers in CAT(0) spaces has been an influential tool to study the geometric mean or also called Karcher barycenter of positive definite matrices. It provides an easily computable stochastic approximation based on inductive means. Convergence of a deterministic version of this approximation has been proved by Holbrook, providing his "nodice" theorem for the Karcher mean of positive definite matrices. The Karcher mean has also been extended to the infinite dimensional case of positive operators on a Hilbert space by Lawson-Lim and then to probability measures with bounded support by the second author, however the CAT(0) property of the space is lost and one defines the mean as the unique solution of a nonlinear operator equation on a convex Banach-Finsler manifold. The formulations of Sturm's strong law of large numbers and Holbrook's "nodice" approximation are natural and both conjectured to converge, however all previous techniques of their proofs break down, due to the Banach-Finsler nature of the space. In this paper we prove both conjectures by establishing the most general L 1 -form of Sturm's strong law of large numbers and Holbrook's "nodice" theorem in the operator norm by developing a stochastic discrete-time resolvent flow for the Karcher barycenter using its Wasserstein contraction property.

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Gradient flows and a Trotter-Kato formula of semi-convex functions on CAT(1)-spaces

Cited by 21 publications

References 30 publications

A Differential Perspective on Gradient Flows on $$\textsf {CAT} (\kappa )$$-Spaces and Applications

A Differential Perspective on Gradient Flows on $$\textsf {CAT} (\kappa )$$-Spaces and Applications

Improvements of upper curvature bounds

Strong law of large numbers for the $L^1$-Karcher mean

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