Modified log-Sobolev inequalities, Beckner inequalities and moment estimates

Modified Log-sobolev Inequalities and Isoperimetry

We find sufficient conditions for a probability measure μ to satisfy an inequality of the type ∫ Rd fF ( f ∫ Rd f 2 dμ ) dμ ≤ C ∫ Rd f2c∗ ( |∇f | |f | ) dμ + B ∫ Rd f dμ, where F is concave and c (a cost function) is convex. We show that under broad assumptions on c and F the above inequality holds if for some δ > 0 and ε > 0 one has ∫ ε 0 Φ ( δc [ tF ( 1t ) Iμ(t) ]) dt <∞, where Iμ is the isop...

Modified logarithmic Sobolev inequalities and transportation inequalities

Modified Logarithmic Sobolev Inequalities in Discrete Settings

Motivated by the rate at which the entropy of an ergodic Markov chain relative to its stationary distribution decays to zero, we study modified versions of logarithmic Sobolev inequalities in the discrete setting of finite Markov chains and graphs. These inequalities turn out to be weaker than the standard log-Sobolev inequality, but stronger than the Poincare’ (spectral gap) inequality. We sho...

Modified logarithmic Sobolev inequalities in null curvature

We present a logarithmic Sobolev inequality adapted to a log-concave measure. Assume that Φ is a symmetric convex function on R satisfying (1 + ε)Φ(x) 6 xΦ(x) 6 (2 − ε)Φ(x) for x > 0 large enough and with ε ∈]0, 1/2]. We prove that the probability measure on R μΦ(dx) = e /ZΦdx satisfies a modified and adapted logarithmic Sobolev inequality : there exist three constant A,B,D > 0 such that for al...

Hypercontractivity and Log Sobolev Inequalities in Quantum Information Theory (15w5098)

S (in alphabetic order by speaker surname) Speaker: William Beckner (University of Texas at Austin) Title: Hypercontractivity and the Logarithmic Sobolev Inequality – from Quantum Field Theory to Geometric Inequalities Abstract: This talk will feature a quick historical survey of connections between the log Sobolev inequality and now classical problems in analysis, physics and probability. Spea...