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Entropy bounds on abelian groups and the ruzsa divergence
Over the past few years, a family of interesting new inequalities for the
entropies of sums and differences of random variables has been developed by
Ruzsa, Tao and others, motivated by analogous results in additive
combinatorics. The present work extends these earlier results to the case of
random variables taking values in or, more generally, in
arbitrary locally compact and Polish abelian groups. We isolate and study a key
quantity, the Ruzsa divergence between two probability distributions, and we
show that its properties can be used to extend the earlier inequalities to the
present general setting. The new results established include several variations
on the theme that the entropies of the sum and the difference of two
independent random variables severely constrain each other. Although the
setting is quite general, the result are already of interest (and new) for
random vectors in . In that special case, quantitative bounds are
provided for the stability of the equality conditions in the entropy power
inequality; a reverse entropy power inequality for log-concave random vectors
is proved; an information-theoretic analog of the Rogers-Shephard inequality
for convex bodies is established; and it is observed that some of these results
lead to new inequalities for the determinants of positive-definite matrices.
Moreover, by considering the multiplicative subgroups of the complex plane, one
obtains new inequalities for the differential entropies of products and ratios
of nonzero, complex-valued random variables
Stability for a GNS inequality and the Log-HLS inequality, with application to the critical mass Keller-Segel equation
Starting from the quantitative stability result of Bianchi and Egnell for the
2-Sobolev inequality, we deduce several different stability results for a
Gagliardo-Nirenberg-Sobolev inequality in the plane. Then, exploiting the
connection between this inequality and a fast diffusion equation, we get a
quantitative stability for the Log-HLS inequality. Finally, using all these
estimates, we prove a quantitative convergence result for the critical mass
Keller-Segel system
Stability results for logarithmic Sobolev and Gagliardo-Nirenberg inequalities
This paper is devoted to improvements of functional inequalities based on
scalings and written in terms of relative entropies. When scales are taken into
account and second moments fixed accordingly, deficit functionals provide
explicit stability measurements, i.e., bound with explicit constants distances
to the manifold of optimal functions. Various results are obtained for the
Gaussian logarithmic Sobolev inequality and its Euclidean counterpart, for the
Gaussian generalized Poincar{\'e} inequalities and for the Gagliardo-Nirenberg
inequalities. As a consequence, faster convergence rates in diffusion equations
(fast diffusion, Ornstein-Uhlenbeck and porous medium equations) are obtained
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