Logarithmic Sobolev Inequalities









Logarithmic Sobolev Inequalities

logarithmic Sobolev inequalities what they are some history analytic
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Logarithmic Sobolev Inequalities

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213515 Logarithmic Sobolev Inequalities

Logarithmic Sobolev Inequalities

M. Ledoux

Institut de Mathematiques de Toulouse, France

logarithmic Sobolev inequalities what they are, some history analytic, geometric, optimal transportation proofs last decade developments at the interface b etween analysis, probability, geometry what are logarithmic Sobolev inequalities?

Sobolev inequalities

f:Rm!Rsmooth, compactly supported Z R mjfjpdx 2=p CmZ R mjrfj2dx p=2mm2(>2) (m3) sharp constantCm=1m(m2) (m)( m2 2=m Z R mjfjpdx 2=p CmZ R mjrfj2dx 2p log Z R mjfjpdx log C mZ R mjrfj2dx assume Z R mf2dx= 1

Jensen's inequality

fo rf2dx log Z R mjfjpdx = log Z R mjfjp2f2dx Z R mlogjfjp2f2dx p2pZ R mf2logf2dxlog C mZ R mjrfj2dx p2p Z R mf2logf2dxlog C mZ R mjrfj2dx ;Z R mf2dx= 1 form of loga rithmicSob olevinequalit y formally come back to

Sob olev(w orseconstan ts)

issue on sha rpconstants f:Rn!Rsmooth,R R nf2dx= 1 f kn:Rkn!R;m=kn;k! 1 Z R nf2logf2dxn2 log2neZ R njrfj2dx ;Z R nf2dx= 1 sharp (Euclidean)logarithmic Sobolev inequality used byG. Perelman (2002) (Euclidean)logarithmic Sobolev inequality Z R nf2logf2dxn2 log2neZ R njrfj2dx ;Z R nf2dx= 1 dx!d(x) =ejxj2=2dx(2)n=2 standard Gaussian probability measure onRn changef2intof2ejxj2=2 f:Rn!Rsmooth,R R nf2d= 1 Z R nf2logf2d2Z R njrfj2d (Gaussian)logarithmic Sobolev inequality Z R nf2logf2d2Z R njrfj2d;Z R nf2d= 1 d(x) =ejxj2=2dx(2)n=2

Sobolev type inequality

(fo r) constant is sharp constant independent of n(stabilityb yp roduct) extension to innite dimensional Wiener space

Gibbs measures, models from statistical mechanics

(Gaussian)logarithmic Sobolev inequality Z R nf2logf2d2Z R njrfj2d;Z R nf2d= 1 d(x) =ejxj2=2dx(2)n=2 dierent forms

Sobolev type inequality

information theory

PDE formulation

information theory description f!pf;f>0;R R nf d= 1 Z R nflogf d12 Z R njrfj2f d d=f dprobability Z R nflogf d=Hj(relative)entropy Z R njrfj2f d=Ij(relative)Fisher information entropyHj12

IjFisher information

PDE description

ffunction!probability (Lebesgue) density

1=ejxj2=2(2)n=2

R R nf d= 1; =f1;R R

Logarithmic Sobolev Inequalities

M. Ledoux

Institut de Mathematiques de Toulouse, France

logarithmic Sobolev inequalities what they are, some history analytic, geometric, optimal transportation proofs last decade developments at the interface b etween analysis, probability, geometry what are logarithmic Sobolev inequalities?

Sobolev inequalities

f:Rm!Rsmooth, compactly supported Z R mjfjpdx 2=p CmZ R mjrfj2dx p=2mm2(>2) (m3) sharp constantCm=1m(m2) (m)( m2 2=m Z R mjfjpdx 2=p CmZ R mjrfj2dx 2p log Z R mjfjpdx log C mZ R mjrfj2dx assume Z R mf2dx= 1

Jensen's inequality

fo rf2dx log Z R mjfjpdx = log Z R mjfjp2f2dx Z R mlogjfjp2f2dx p2pZ R mf2logf2dxlog C mZ R mjrfj2dx p2p Z R mf2logf2dxlog C mZ R mjrfj2dx ;Z R mf2dx= 1 form of loga rithmicSob olevinequalit y formally come back to

Sob olev(w orseconstan ts)

issue on sha rpconstants f:Rn!Rsmooth,R R nf2dx= 1 f kn:Rkn!R;m=kn;k! 1 Z R nf2logf2dxn2 log2neZ R njrfj2dx ;Z R nf2dx= 1 sharp (Euclidean)logarithmic Sobolev inequality used byG. Perelman (2002) (Euclidean)logarithmic Sobolev inequality Z R nf2logf2dxn2 log2neZ R njrfj2dx ;Z R nf2dx= 1 dx!d(x) =ejxj2=2dx(2)n=2 standard Gaussian probability measure onRn changef2intof2ejxj2=2 f:Rn!Rsmooth,R R nf2d= 1 Z R nf2logf2d2Z R njrfj2d (Gaussian)logarithmic Sobolev inequality Z R nf2logf2d2Z R njrfj2d;Z R nf2d= 1 d(x) =ejxj2=2dx(2)n=2

Sobolev type inequality

(fo r) constant is sharp constant independent of n(stabilityb yp roduct) extension to innite dimensional Wiener space

Gibbs measures, models from statistical mechanics

(Gaussian)logarithmic Sobolev inequality Z R nf2logf2d2Z R njrfj2d;Z R nf2d= 1 d(x) =ejxj2=2dx(2)n=2 dierent forms

Sobolev type inequality

information theory

PDE formulation

information theory description f!pf;f>0;R R nf d= 1 Z R nflogf d12 Z R njrfj2f d d=f dprobability Z R nflogf d=Hj(relative)entropy Z R njrfj2f d=Ij(relative)Fisher information entropyHj12

IjFisher information

PDE description

ffunction!probability (Lebesgue) density

1=ejxj2=2(2)n=2

R R nf d= 1; =f1;R R
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