Functional inequalities for the heat flow on time‐dependent metric measure spaces

Kopfer, Eva; Sturm, Karl-Theodor

doi:10.1112/jlms.12452

Cited by 4 publications

(2 citation statements)

References 22 publications

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“…We currently do not know of any interesting applications of the generalised set-up of Theorem 3.20 over that of Theorem 3.6, but it would be very interesting to find some. Some references with related constructions (though different motivation) in the context of the Ricci flow appeared in [77] and then [65,70] and more recently [56,59,60,87].…”

Section: Aside: Geometric Perspective On the Polchinski Flowmentioning

confidence: 99%

Spectral Gap Critical Exponent for Glauber Dynamics of Hierarchical Spin Models

Bauerschmidt

Bodineau

2019

Commun. Math. Phys.

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We develop a renormalisation group approach to deriving the asymptotics of the spectral gap of the generator of Glauber type dynamics of spin systems with strong correlations (at and near a critical point). In our approach, we derive a spectral gap inequality for the measure recursively in terms of spectral gap inequalities for a sequence of renormalised measures. We apply our method to hierarchical versions of the 4-dimensional n-component |ϕ| 4 model at the critical point and its approach from the high temperature side, and of the 2-dimensional Sine-Gordon and the Discrete Gaussian models in the rough phase (Kosterlitz-Thouless phase). For these models, we show that the spectral gap decays polynomially like the spectral gap of the dynamics of a free field (with a logarithmic correction for the |ϕ| 4 model), the scaling limit of these models in equilibrium.

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Section: Aside: Geometric Perspective On the Polchinski Flowmentioning

confidence: 99%

Spectral Gap Critical Exponent for Glauber Dynamics of Hierarchical Spin Models

Bauerschmidt

Bodineau

2019

Commun. Math. Phys.

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“…estimates for volume growth and diameter, gradient estimates, transport estimates, Harnack inequalities, logarithmic Sobolev inequalities, isoperimetric inequalities, splitting theorems, maximal diameter theorems, and further rigidity results, see e.g. [2,28,12,19,26,27,17] and references therein. Moreover, deep results on the local structure of metric measure spaces with synthetic Ricci bounds have been obtained recently [32], [11] and an impressive second order calculus has been developed [21].…”

Section: Introductionmentioning

confidence: 99%

Tamed spaces -- Dirichlet spaces with distribution-valued Ricci bounds

Erbar¹,

Rigoni²,

Sturm³

et al. 2020

Preprint

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We develop the theory of tamed spaces which are Dirichlet spaces with distributionvalued lower bounds on the Ricci curvature and investigate these from an Eulerian point of view. To this end we analyze in detail singular perturbations of Dirichlet form by a broad class of distributions. The distributional Ricci bound is then formulated in terms of an integrated version of the Bochner inequality using the perturbed energy form and generalizing the well-known Bakry-Émery curvature-dimension condition. Among other things we show the equivalence of distributional Ricci bounds to gradient estimates for the heat semigroup in terms of the Feynman-Kac semigroup induced by the taming distribution as well as consequences in terms of functional inequalities. We give many examples of tamed spaces including in particular Riemannian manifolds with either interior singularities or singular boundary behavior. MATTHIAS ERBAR, CHIARA RIGONI, KARL-THEODOR STURM, AND LUCA TAMANINI 4.2. A manifold which is tamed but not 2-tamed 36 4.3. Manifolds with boundary and potentially singular curvature 36 4.4. A tamed domain with boundary that is not semiconvex 40 4.5. A Tamed Manifold with Highly Irregular Boundary 41 5. Functional Inequalities for Tamed Spaces 45 6. Self-Improvement of the Taming Condition 47 6.1. Measure-Valued Taming Operator and Bochner Inequality 47 6.2. Self-Improvement of the L 2 -Taming Condition 51 7. Sub-tamed Spaces 55 7.1. Reflected Dirichlet Spaces and Sub-taming 56 7.2. Doubling of Dirichlet Spaces and Sub-taming 58 7.3. Doubling of Riemannian Surfaces 64 References 66

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