Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSelf-improvement of the Bakry-Emery criterion for Poincar{e} inequalities and Wasserstein contraction using variable curvature bounds
201
0
0.0
(
0
)
Authors
Patrick Cattiaux
Ask ChatGPT about the research
No Arabic abstract
We study Poincar{e} inequalities and long-time behavior for diffusion processes on R^n under a variable curvature lower bound, in the sense of Bakry-Emery. We derive various estimates on the rate of convergence to equilibrium in L^1 optimal transport distance, as well as bounds on the constant in the Poincar{e} inequality in several situations of interest, including some where curvature may be negative. In particular, we prove a self-improvement of the Bakry-Emery estimate for Poincar{e} inequalities when curvature is positive but not constant.
rate research
Read More
If Poincar{e} inequality has been studied by Bobkov for radial measures, few is known about the logarithmic Sobolev inequalty in the radial case. We try to fill this gap here using different methods: Bobkovs argument and super-Poincar{e} inequalities, direct approach via L1-logarithmic Sobolev inequalities. We also give various examples where the obtained bounds are quite sharp. Recent bounds obtained by Lee-Vempala in the logconcave bounded case are refined for radial measures.
For Riemannian manifolds with a smooth measure $(M, g, e^{-f}dv_{g})$, we prove a generalized Myers compactness theorem when Bakry--Emery Ricci tensor is bounded from below and $f$ is bounded.
We demonstrate lower bounds for the eigenvalues of compact Bakry-Emery manifolds with and without boundary. The lower bounds for the first eigenvalue rely on a generalised maximum principle which allows gradient estimates in the Riemannian setting to be directly applied to the Bakry-Emery setting. Lower bounds for all eigenvalues are demonstrated using heat kernel estimates and a suitable Sobolev inequality.
Following the recently obtained complete classification of quantum-deformed $mathfrak{o}(4)$, $mathfrak{o}(3,1)$ and $mathfrak{o}(2,2)$ algebras, characterized by classical $r$-matrices, we study their inhomogeneous $D = 3$ quantum IW contractions (i.e. the limit of vanishing cosmological constant), with Euclidean or Lorentzian signature. Subsequently, we compare our results with the complete list of $D = 3$ inhomogeneous Euclidean and $D = 3$ Poincar{e} quantum deformations obtained by P.~Stachura. It turns out that the IW contractions allow us to recover all Stachura deformations. We further discuss the applicability of our results in the models of 3D quantum gravity in the Chern-Simons formulation (both with and without the cosmological constant), where it is known that the relevant quantum deformations should satisfy the Fock-Rosly conditions. The latter deformations in part of the cases are associated with the Drinfeld double structures, which also have been recently investigated in detail.
For any bounded domain $Omega$ in $mathbb C^m,$ let ${mathrm B}_1(Omega)$ denote the Cowen-Douglas class of commuting $m$-tuples of bounded linear operators. For an $m$-tuple $boldsymbol T$ in the Cowen-Douglas class ${mathrm B}_1(Omega),$ let $N_{boldsymbol T}(w)$ denote the restriction of $boldsymbol T$ to the subspace ${cap_{i,j=1}^mker(T_i-w_iI)(T_j-w_jI)}.$ This commuting $m$-tuple $N_{boldsymbol T}(w)$ of $m+1$ dimensional operators induces a homomorphism $rho_{_{!N_{boldsymbol T}(w)}}$ of the polynomial ring $P[z_1, ..., z_m],$ namely, $rho_{_{!N_{boldsymbol T}(w)}}(p) = pbig (N_{boldsymbol T}(w) big),, pin P[z_1, ..., z_m].$ We study the contractivity and complete contractivity of the homomorphism $rho_{_{!N_{boldsymbol T}(w)}}.$ Starting from the homomorphism $rho_{_{!N_{boldsymbol T}(w)}},$ we construct a natural class of homomorphism $rho_{_{!N^{(lambda)}(w)}}, lambda>0,$ and relate the properties of $rho_{_{!N^{(lambda)}(w)}}$ to that of $rho_{_{!N_{boldsymbol T}(w)}}.$ Explicit examples arising from the multiplication operators on the Bergman space of $Omega$ are investigated in detail. Finally, it is shown that contractive properties of $rho_{_{!N_{boldsymbol T}(w)}}$ is equivalent to an inequality for the curvature of the Cowen-Douglas bundle $E_{boldsymbol T}$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
