{"id":16672,"date":"2023-01-12T19:08:07","date_gmt":"2023-01-12T18:08:07","guid":{"rendered":"https:\/\/djalil.chafai.net\/blog\/?p=16672"},"modified":"2024-04-01T20:35:43","modified_gmt":"2024-04-01T18:35:43","slug":"log-sobolev-and-bakry-emery","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2023\/01\/12\/log-sobolev-and-bakry-emery\/","title":{"rendered":"Log-Sobolev and Bakry-\u00c9mery"},"content":{"rendered":"
\"Leonard<\/a>
Leonard Gross (1931 -- )<\/figcaption><\/figure>\n

This post is formed with the rough notes that I have prepared for a long informal talk given on January 3, 2023, at Paris-Dauphine, around log-Sobolev inequalities and the Bakry-\u00c9mery criterion. Many aspects are already in Master 2 Lecture Notes<\/a> written with Joseph Lehec.<\/p>\n

The logarithmic Sobolev inequality (LSI) concept was forged by Leonard Gross<\/a> (1931 -) in 1975<\/a>, as a reformulation of the hypercontractivity of a Markov semigroup. More precisely, if $(P_t)_{t\\geq0}=(\\mathrm{e}^{tL})_{t\\geq}$ is a Markov semigroup with infinitesimal generator $L$ and invariant probability measure $\\mu$, and if $L$ is a diffusion or if $\\mu$ is reversible, then for all constant $c>0$,
\n\\[
\n\\|P_t(f)\\|_{1+(p-1)\\mathrm{e}^{4t\/c}}\\leq\\|f\\|_p,\\quad \\forall t\\geq0,\\forall p\\geq1,\\forall f,
\n\\] if and only if
\n\\[
\n\\int f^2\\log(f^2)\\mathrm{d}\\mu
\n\\leq -c\\int fLf\\mathrm{d}\\mu+\\int f^2\\mathrm{d}\\mu\\log\\int f^2\\mathrm{d}\\mu\\,\\quad\\forall f.
\n\\] The name hypercontractivity comes from the fact that $1+(p-1)\\mathrm{e}^{4t\/c}>p$ if $t>0$. The name LSI comes from an analogy with classical Sobolev inequalities. The logarithm in the LSI comes from hypercontractivity as a derivative of the $L^p$ norm with respect to $p$.
\n\\[
\n\\partial_p\\int |f|^p\\mathrm{d}\\mu=\\int|f|^p\\log|f|\\mathrm{d}\\mu.
\n\\] The assumption of being a diffusion or reversible allows to transform the LSI into a $p$-homogeneous statement by a simple power change of function. The hypercontractivity inequality is an equality at time $t=0$, and the LSI is its infinitesimal version, via $L=\\partial_{t=0}P_t$. The same holds for all $t$ due to the Markov nature of the semigroup and the invariance of $\\mu$.<\/p>\n

Still in the reversible Markovian context, the LSI is also equivalent to the sub-exponential decay in time of relative entropy along the dynamics, namely, with $f_t:=P_t(f)$, $f\\geq0$, $\\int f\\mathrm{d}\\mu=1$ :
\n$$
\n\\int f_t\\log(f_t)\\mathrm{d}\\mu
\n\\leq\\mathrm{e}^{-4t\/c}\\int f_0\\log(f_0)\\mathrm{d}\\mu,\\quad\\forall f\\geq0,\\forall t\\geq0.
\n$$ The Boltzmannian H-theorem for the (linear) evolution equation $\\partial_tf_t=Lf_t$ reads
\n$$\\partial_t\\int f_t\\log(f_t)\\mathrm{d}\\mu=\\int\\log(f_t)Lf_t\\mathrm{d}t\\leq0$$ while the sub-exponential decay above is the corresponding Cercignany theorem. In this context, and beyond the monotonicity, the Bakry-\u00c9mery criterion provides a convexity in time, as well as the exponential decay via a Gr\u00f6nwall lemma. It is this connection with kinetic theory and the Boltzmann PDE that brought C\u00e9dric Villani to the domain in the late 1990s.<\/p>\n

We know nowadays that the LSI is linked with information theory, functional analysis, analysis on manifolds, statistical mechanics, harmonic analysis, analysis of PDE, stochastic processes, free probability, high dimensional probability, high dimensional statistics, among other fields. Inspired by C\u00e9dric Villani, a historical note by Michel Ledoux gathers 15 proofs of the Gaussian LSI<\/a>.<\/p>\n

Leonard Gross should not be confused with the physicist David J. Gross<\/a> (1941 -- ), the theoretical physicist Eugene P. Gross<\/a> (1926 -- 1991), the mathematicians Benedict Hyman Gross<\/a> (1950 -- ) and Mark Gross<\/a> (1965 - his son!), among other big Gross<\/a>.<\/p>\n

The Gaussian LSI was already studied, in its Lebesgue form in 1959<\/a> by the information theorist Aart Johannes Stam<\/a> (1929 -- 2020), in 1969<\/a> by the mathematical physicist Paul Gerard Federbush<\/a> (1934 -- ), an academic grandson of Enrico Fermi<\/a>, and the academic grandfather of Roland Bauerschmidt<\/a>, and in 1975<\/a> by William G. Faris<\/a> (1939 -- ).<\/p>\n

In the 1980s, the LSI and related functional inequalities were also studied by Paul-Andr\u00e9 Meyer<\/a> and Dominique Bakry<\/a>, Michel \u00c9mery<\/a>, as well as Michel Ledoux<\/a>, Daniel Stroock<\/a>, Oscar Rothaus<\/a>, ... In the 1990s, came Laurent Miclo<\/a>, William Beckner<\/a>, Laurent Saloff-Coste<\/a>, Persi Diaconis<\/a>, ..., but also, for statistical mechanics, Horng-Tzer Yau<\/a>, Boguslaw Zegarlinski<\/a>, Fabio Martinelli<\/a>, Thierry Bodineau<\/a>, Bernard Helffer<\/a>, ... In the 2000s, came Sergey Bobkov<\/a>, C\u00e9dric Villani<\/a>, Liming Wu<\/a>, Patrick Cattiaux<\/a>, Arnaud Guillin<\/a>,...<\/p>\n

\"Paul<\/a>
Paul Gerard Federbush (1934 -- )<\/figcaption><\/figure>\n

Variance, entropies, Fisher.<\/strong>
\n\\begin{align*}
\n\\mathrm{Var}_{\\mu}(f)&=\\int f^2\\mathrm{d}\\mu-\\left(\\int f\\mathrm{d}\\mu\\right)^2=\\mathrm{Var}(f(X))\\quad\\text{where $X\\sim\\mu$}\\\\
\n\\mathrm{Ent}_{\\mu} (f)&=\\int f \\log f\\mathrm{d}\\mu - \\left( \\int f\\mathrm{d}\\mu \\right)\\log \\left( \\int f\\mathrm{d}\\mu \\right),\\quad f\\geq0\\\\
\n\\mathrm{Ent}_{\\mu}^\\Phi(f)&=\\int\\Phi(f)\\mathrm{d}\\mu-\\Phi\\Bigr(\\int f\\mathrm{d}\\mu\\Bigr) =\\mathbb{E}(\\Phi(f(X)))-\\Phi(\\mathbb{E}(f(X))\\\\
\n\\Phi(u)&=u^2, u\\in\\mathbb{R}\\\\
\n\\Phi(u)&=u\\log(u), u\\geq0
\n\\end{align*} To make it $2$-homogeneous like the variance : $\\mathrm{Ent}_{\\mu}(f^2)$
\nKullback-Leibler divergence (relative entropy) : $\\mathrm{H}(\\nu\\mid\\mu)\\geq0$, $\\mathrm{d}\\nu:=f\\mathrm{d}\\mu$
\nOn $\\mathbb{R}^n$ relation to statistical physics\/mechanics :<\/p>\n