Horizontal Diffusion in C 1 Path Space

Arnaudon, Marc; Coulibaly, Kolehe Abdoulaye; Thalmaier, Anton

doi:10.1007/978-3-642-15217-7_2

Cited by 28 publications

(53 citation statements)

References 21 publications

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“…It will be quite helpful for the study of Ricci flow on a noncompact manifold by means of the heat equation. In fact, our result enables us to remove the assumption on the non-explosion in recent work by Arnaudon et al [2,Sect. 4].…”

Section: Remarks Concerning Related Workmentioning

confidence: 99%

“…They extend McCann and Topping's implication 1 ⇒ 2 in the case on a noncompact manifold. In addition, they sharpen the monotonicity of L 2 -Wasserstein distance to a pathwise contraction in the following sense; there is a coupling (X (1) t ,X (2) t ) t≥0 of two Brownian motions starting from x, y ∈ X respectively so that t → d g(t) (X (1) t ,X (2) t ) is non-increasing almost surely. By taking an expectation, we can derive the monotonicity of the L 2 -Wasserstein distance from it.…”

Section: Remarks Concerning Related Workmentioning

confidence: 99%

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Non-explosion of diffusion processes on manifolds with time-dependent metric

Kuwada

Philipowski

2010

Math. Z.

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We study the problem of non-explosion of diffusion processes on a manifold with time-dependent Riemannian metric. In particular we obtain that Brownian motion cannot explode in finite time if the metric evolves under backwards Ricci flow. Our result makes it possible to remove the assumption of non-explosion in the pathwise contraction result established by Arnaudon et al. (arXiv:0904.2762, to appear in Sém. Prob.). As an important tool which is of independent interest we derive an Itô formula for the distance from a fixed reference point, generalising a result of Kendall (Ann. Prob. 15:1491-1500, 1987.Keywords Ricci flow · Diffusion process · Non-explosion · Radial process Mathematics Subject Classification (2000) 53C21 · 53C44 · 58J35 · 60J60 Brownian motion with respect to time-changing Riemannian metricsLet M be a d-dimensional differentiable manifold with d ≥ 2, π : F (M) → M the frame bundle and (g(t)) t∈[0,T ] a family of Riemannian metrics on M depending smoothly on t such that (M, g(t)) is geodesically complete for all tβ=1 be the canonical vertical vector fields. Let (W t ) t≥0 be a standard R d -valued Brownian motion. In this situation Arnaudon et al. [1,5] defined horizontal Brownian motion on F (M) as the solution of the K. Kuwada was partially supported by the JSPS fellowship for research abroad.

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Section: Remarks Concerning Related Workmentioning

confidence: 99%

Section: Remarks Concerning Related Workmentioning

confidence: 99%

Non-explosion of diffusion processes on manifolds with time-dependent metric

Kuwada

Philipowski

2010

Math. Z.

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“…It was since extended to handle the W p distance for arbitrary p [1]. The only other case relevant to us is the case p = 1 (i.e.…”

Section: Contractivity For Diffusions On Evolving Manifoldsmentioning

confidence: 99%

Ricci flow: the foundations via optimal transportation

Topping¹

2014

Optimal Transport

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“…The original intuition [Vey-a] was to couple two Brownian motions starting at very close points and using curvature, an approach used in [Ken86] (see [ACT11] for a new elegant viewpoint). If Brownian motion is seen as a large number of small steps, using parallel transport of the trajectory (x t ) starting at x provides a Brownian trajectory (y t ) starting at y close to x; since, at each step, the distance between x t and y t decreases by a factor depending on Ricci curvature, the distance after time t decreases exponentially fast: Ricci curvature is a kind of "Lyapunov exponent along Brownian trajectories".…”

Section: Nmentioning

confidence: 99%

A visual introduction to Riemannian curvatures and some discrete generalizations

Ollivier¹

2013

CRM Proceedings and Lecture Notes

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Abstract. We try to provide a visual introduction to some objects used in Riemannian geometry: parallel transport, sectional curvature, Ricci curvature, Bianchi identities... We then explain some of the strategies used to define analogues of curvature in non-smooth or discrete spaces, beginning with Alexandrov curvature and δ-hyperbolic spaces, and insisting on various notions of generalized Ricci curvature, which we briefly compare.The first part of this text covers in a hopefully intuitive and visual way some of the usual objects of Riemannian geometry: parallel transport, sectional curvature, Ricci curvature, the Riemann tensor, Bianchi identities... For each of those we try to provide one or several pictures that convey the meaning (or at least one possible interpretation) of the formal definition.Next, we consider the problem of defining analogues of curvature for nonsmooth or discrete objets. This is in the spirit of "synthetic" or "coarse" geometry, in which large-scale properties of metric spaces are investigated instead of fine smallscale properties; one of the aims being to be impervious to small perturbations of the underlying space. Motivation for these non-smooth or discrete extensions often comes from various other fields of mathematics, such as group theory or optimal transport. From this viewpoint is emerging a theory of metric measure spaces, see for instance [Gro99].Thus we cover the definitions and mention some applications of δ-hyperbolic spaces and Alexandrov curvature (generalizing sectional curvature), and of two notions of generalized Ricci curvature: displacement convexity of entropy introduced by Sturm and Lott-Villani (following Renesse-Sturm and others), and coarse Ricci curvature as used by the author. We mention new theorems obtained for the original Riemannian case thanks to this more general viewpoint. We also briefly discuss the differences between these two approaches to Ricci curvature.Note that scalar curvature is not covered, for lack of the author's competence about recent developments and applications in this field.

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Horizontal Diffusion in C 1 Path Space

Cited by 28 publications

References 21 publications

Non-explosion of diffusion processes on manifolds with time-dependent metric

Non-explosion of diffusion processes on manifolds with time-dependent metric

Ricci flow: the foundations via optimal transportation

A visual introduction to Riemannian curvatures and some discrete generalizations

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