arXiv subjects
Publications and source records attributed to Devraj Duggal.
Pinsker-type inequalities are considered for the weighted total variation distance between probability measures in terms of the R\'enyi divergence powers. They are applied in derivation of transport-entropy inequalities under moment-type conditions.
We develop a general comparison result for inf convolution operators related to rearrangement. As a consequence we derive comparison results under spherically symmetric rearrangement for Laplace and polar transforms. As a by product we simplify existing proofs related to the functional Blaschke Santalo inequality of Keith Ball and derive a comparison result for some parabolic PDE.
Covariance representations are developed for the uniform distributions on the Euclidean spheres in terms of spherical gradients and Hessians. They are applied to derive a number of Sobolev type inequalities and to recover and refine the concentration of measure phenomenon, including second order concentration inequalities. A detail account is also given in the case of the circle, with a short overview of H\"offding's kernels and covariance identities in the class of periodic functions.
We start with a brief survey on H\"offding's kernels, its properties, related spectral decompositions, and discuss marginal distributions of H\"offding measures. In the second part of this note, one-dimensional covariance representations are considered over compactly supported probability distributions in the class of periodic smooth functions. H\"offding's kernels are used in the construction of mixing measures whose marginals are multiples of given probability distributions, leading to optimal kernels in periodic covariance representations.
Inspired by the approach of Ivanisvili and Volberg towards functional inequalities for probability measures with strictly convex potentials, we investigate the relationship between curvature bounds in the sense of Bakry-Emery and local functional inequalities. We will show that not only is the earlier approach for strictly convex potentials extendable to Markov semigroups and simplified through use of the $\Gamma$-calculus, providing a consolidating machinery for obtaining functional inequalities new and old in this general setting, but that a converse also holds. Local inequalities obtained are proven equivalent to Bakry-Emery curvature. Moreover we will develop this technique for metric measure spaces satisfying the RCD condition, providing a unified approach to functional and isoperimetric inequalities in non-smooth spaces with a synthetic Ricci curvature bound. Finally, we are interested in commutation properties for semi-group operators on $\mathbb{R}^n$ in the absence of positive curvature, based on a local eigenvalue criteria.