1,707 research outputs found
A novel type of Sobolev-Poincar\'e inequality for submanifolds of Euclidean space
For functions on generalised connected surfaces (of any dimensions) with
boundary and mean curvature, we establish an oscillation estimate in which the
mean curvature enters in a novel way. As application we prove an a priori
estimate of the geodesic diameter of compact connected smooth immersions in
terms of their boundary data and mean curvature. These results are developed in
the framework of varifolds. For this purpose, we establish that the notion of
indecomposability is the appropriate substitute for connectedness and that it
has a strong regularising effect; we thus obtain a new natural class of
varifolds to study. Finally, our development leads to a variety of questions
that are of substance both in the smooth and the nonsmooth setting.Comment: 35 pages, no figure
On reverse hypercontractivity
We study the notion of reverse hypercontractivity. We show that reverse
hypercontractive inequalities are implied by standard hypercontractive
inequalities as well as by the modified log-Sobolev inequality. Our proof is
based on a new comparison lemma for Dirichlet forms and an extension of the
Strook-Varapolos inequality.
A consequence of our analysis is that {\em all} simple operators L=Id-\E as
well as their tensors satisfy uniform reverse hypercontractive inequalities.
That is, for all and every positive valued function for we have . This should
be contrasted with the case of hypercontractive inequalities for simple
operators where is known to depend not only on and but also on the
underlying space.
The new reverse hypercontractive inequalities established here imply new
mixing and isoperimetric results for short random walks in product spaces, for
certain card-shufflings, for Glauber dynamics in high-temperatures spin systems
as well as for queueing processes. The inequalities further imply a
quantitative Arrow impossibility theorem for general product distributions and
inverse polynomial bounds in the number of players for the non-interactive
correlation distillation problem with -sided dice.Comment: Final revision. Incorporates referee's comments. The proof of
appendix B has been corrected. A shorter version of this article will appear
in GAF
Upper bounds for the density of solutions of stochastic differential equations driven by fractional Brownian motions
In this paper we study upper bounds for the density of solution of stochastic
differential equations driven by a fractional Brownian motion with Hurst
parameter H > 1/3. We show that under some geometric conditions, in the regular
case H > 1/2, the density of the solution satisfy the log-Sobolev inequality,
the Gaussian concentration inequality and admits an upper Gaussian bound. In
the rough case H > 1/3 and under the same geometric conditions, we show that
the density of the solution is smooth and admits an upper sub-Gaussian bound
Bounds on the negative eigenvalues of Laplacians on finite metric graphs
For a self--adjoint Laplace operator on a finite, not necessarily compact,
metric graph lower and upper bounds on each of the negative eigenvalues are
derived. For compact finite metric graphs Poincar\'{e} type inequalities are
given.Comment: 17 page
From Poincaré Inequalities to Nonlinear Matrix Concentration
This paper deduces exponential matrix concentration from a Poincaré inequality via a short, conceptual argument. Among other examples, this theory applies to matrix-valued functions of a uniformly log-concave random vector. The proof relies on the subadditivity of Poincaré inequalities and a chain rule inequality for the trace of the matrix Dirichlet form. It also uses a symmetrization technique to avoid difficulties associated with a direct extension of the classic scalar argument
- …