The subelliptic heat kernel on SU(2): Representations, Asymptotics and Gradient bounds

The Lie group SU(2) endowed with its canonical subriemannian structure appears as a three-dimensional model of a positively curved subelliptic space. The goal of this work is to study the subelliptic heat kernel on it and some related functional inequalities.Comment: Update: Added section + Correction of typo

Propagation of chaos for rank-based interacting diffusions and long time behaviour of a scalar quasilinear parabolic equation

We study a quasilinear parabolic Cauchy problem with a cumulative distribution function on the real line as an initial condition. We call 'probabilistic solution' a weak solution which remains a cumulative distribution function at all times. We prove the uniqueness of such a solution and we deduce the existence from a propagation of chaos result on a system of scalar diffusion processes, the interactions of which only depend on their ranking. We then investigate the long time behaviour of the solution. Using a probabilistic argument and under weak assumptions, we show that the flow of the Wasserstein distance between two solutions is contractive. Under more stringent conditions ensuring the regularity of the probabilistic solutions, we finally derive an explicit formula for the time derivative of the flow and we deduce the convergence of solutions to equilibrium.Comment: Stochastic partial differential equations: analysis and computations (2013) http://dx.doi.org/10.1007/s40072-013-0014-

Generalized Ricci Curvature Bounds for Three Dimensional Contact Subriemannian manifolds

Measure contraction property is one of the possible generalizations of Ricci curvature bound to more general metric measure spaces. In this paper, we discover sufficient conditions for a three dimensional contact subriemannian manifold to satisfy this property.Comment: 49 page

Transgenic Rescue of the LARGEmyd Mouse: A LARGE Therapeutic Window?

LARGE is a glycosyltransferase involved in glycosylation of α-dystroglycan (α-DG). Absence of this protein in the LARGEmyd mouse results in α-DG hypoglycosylation, and is associated with central nervous system abnormalities and progressive muscular dystrophy. Up-regulation of LARGE has previously been proposed as a therapy for the secondary dystroglycanopathies: overexpression in cells compensates for defects in multiple dystroglycanopathy genes. Counterintuitively, LARGE overexpression in an FKRP-deficient mouse exacerbates pathology, suggesting that modulation of α-DG glycosylation requires further investigation. Here we demonstrate that transgenic expression of human LARGE (LARGE-LV5) in the LARGEmyd mouse restores α-DG glycosylation (with marked hyperglycosylation in muscle) and that this corrects both the muscle pathology and brain architecture. By quantitative analyses of LARGE transcripts we also here show that levels of transgenic and endogenous LARGE in the brains of transgenic animals are comparable, but that the transgene is markedly overexpressed in heart and particularly skeletal muscle (20–100 fold over endogenous). Our data suggest LARGE overexpression may only be deleterious under a forced regenerative context, such as that resulting from a reduction in FKRP: in the absence of such a defect we show that systemic expression of LARGE can indeed act therapeutically, and that even dramatic LARGE overexpression is well-tolerated in heart and skeletal muscle. Moreover, correction of LARGEmyd brain pathology with only moderate, near-physiological LARGE expression suggests a generous therapeutic window

Ricci curvature of finite Markov chains via convexity of the entropy

We study a new notion of Ricci curvature that applies to Markov chains on discrete spaces. This notion relies on geodesic convexity of the entropy and is analogous to the one introduced by Lott, Sturm, and Villani for geodesic measure spaces. In order to apply to the discrete setting, the role of the Wasserstein metric is taken over by a different metric, having the property that continuous time Markov chains are gradient flows of the entropy. Using this notion of Ricci curvature we prove discrete analogues of fundamental results by Bakry--Emery and Otto--Villani. Furthermore we show that Ricci curvature bounds are preserved under tensorisation. As a special case we obtain the sharp Ricci curvature lower bound for the discrete hypercube.Comment: 39 pages, to appear in Arch. Ration. Mech. Ana

Functional inequalities on manifolds with non-convex boundary

In this article, new curvature conditions are introduced to establish functional inequalities including gradient estimates, Harnack inequalities and transportation-cost inequalities on manifolds with non-convex boundary

A glimpse into the differential topology and geometry of optimal transport

This note exposes the differential topology and geometry underlying some of the basic phenomena of optimal transportation. It surveys basic questions concerning Monge maps and Kantorovich measures: existence and regularity of the former, uniqueness of the latter, and estimates for the dimension of its support, as well as the associated linear programming duality. It shows the answers to these questions concern the differential geometry and topology of the chosen transportation cost. It also establishes new connections --- some heuristic and others rigorous --- based on the properties of the cross-difference of this cost, and its Taylor expansion at the diagonal.Comment: 27 page