47 research outputs found
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