Heat flow regularity, Bismut-Elworthy-Li's derivative formula, and pathwise couplings on Riemannian manifolds with Kato bounded Ricci curvature

Braun, Mathias; Güneysu, Batu

doi:10.48550/arxiv.2001.10297

Cited by 3 publications

(7 citation statements)

References 16 publications

(28 reference statements)

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“…We also could derive a remarkable conservativeness criterion which until recently was not known even in the "classical" setting of spaces with uniform lower Ricci bounds (more precisely, neither for Dirichlet spaces with Ricci bounds in the sense of Bakry-Émery nor for metric measure spaces with Ricci bounds in the sense of Lott-Sturm-Villani). Recently, a similar result has been obtained in [9] for smooth manifolds with Ricci curvature bounded below by a function in the Kato (or more generally Dynkin) class.…”

Section: Dsupporting

confidence: 63%

“…In particular, for k = 2 3 k c the function V is moderate but not 2-moderate. (ii) Similarly, for 9 4 m 2 and not moderate for k > k c .…”

Section: 21mentioning

confidence: 95%

“…(i) "Singularity of Ricci at ∞": Smooth Riemannian manifolds with Ricci curvature bounded from below in terms of a continuous -but unbounded -function which globally lies in the Kato class, see e.g. recent results for such manifolds [35], [9]. (ii) "Local singularities of Ricci": Riemannian manifolds with (synthetic) Ricci curvature bounded from below in terms of a locally unbounded function which lies in L p for some p > n/2.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Dsupporting

confidence: 63%

“…In particular, for k = 2 3 k c the function V is moderate but not 2-moderate. (ii) Similarly, for 9 4 m 2 and not moderate for k > k c .…”

Section: 21mentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

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

Erbar¹,

Rigoni²,

Sturm³

et al. 2020

Preprint

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“…4.4] (recall Example 1.18). More generally [15,18], let M be a "regular" Lipschitz Riemannian manifold, in the sense of [18], that is quasi-isometric to a Riemannian manifold with uniformly lower bounded Ricci curvature. Suppose that the Ricci curvature of M , where defined, is bounded from below by a function k…”

Section: Given Any Borelianmentioning

confidence: 99%

“…Of particular interest in the outlined business of singular Ricci bounds is the extended Kato class K 1− (M ) of signed measures on M , already for Riemannian manifolds without boundary [15,24,49,50,51,79,85,86] or their Ricci limits [25]. This is just the right class of measure-valued potentials for which the associated Feynman-Kac semigroup has good properties [92].…”

Section: Introductionmentioning

confidence: 99%

Vector calculus for tamed Dirichlet spaces

Braun¹

2021

Preprint

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In the language of L ∞ -modules proposed by Gigli, we introduce a first order calculus on a topological Lusin measure space (M , m) carrying a quasi-regular, strongly local Dirichlet form E. Furthermore, we develop a second order calculus if (M , E, m) is tamed by a signed measure in the extended Kato class in the sense of Erbar, Rigoni, Sturm and Tamanini. This allows us to define e.g. Hessians, covariant and exterior derivatives, Ricci curvature, and second fundamental form.

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Heat flow on 1-forms under lower Ricci bounds. Functional inequalities, spectral theory, and heat kernel

Braun¹

2020

Preprint

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We study the canonical heat flow (H ) ≥0 on the cotangent module 2 ( * ) over an RCD( , ∞) space ( , d, m), ∈ ℝ. We show Hess-Schrader-Uhlenbrock's inequality and, if ( , d, m) is also an RCD * ( , ) space, ∈ (1, ∞), Bakry-Ledoux's inequality for (H ) ≥0 w.r.t. the heat flow (P ) ≥0 on 2 ( ). A crucial tool is that the dimensional vector 2-Bochner inequality is self-improving, entailing a dimensional vector 1-Bochner inequality-a version of which is also available in the dimension-free case-as a byproduct. Variable versions of these estimates are discussed as well. In conjunction with a study of logarithmic Sobolev inequalities for 1-forms, the previous inequalities yield various -properties of (H ) ≥0 , ∈ [1, ∞].Then we establish explicit inclusions between the spectrum of its generator, the Hodge Laplacian ⃗ Δ, of the negative functional Laplacian −Δ, and of the Schrödinger operator −Δ + .In the RCD * ( , ) case, we prove compactness of ⃗ Δ −1 if is compact, and the independence of the -spectrum of ⃗ Δ on ∈ [1, ∞] under a volume growth condition.We terminate by giving an appropriate interpretation of a heat kernel for (H ) ≥0 . We show its existence in full generality without any local compactness or doubling assumptions, and derive fundamental estimates and properties of it.

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Heat flow regularity, Bismut-Elworthy-Li's derivative formula, and pathwise couplings on Riemannian manifolds with Kato bounded Ricci curvature

Cited by 3 publications

References 16 publications

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

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

Vector calculus for tamed Dirichlet spaces

Heat flow on 1-forms under lower Ricci bounds. Functional inequalities, spectral theory, and heat kernel

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