logarithmic Sobolev inequalities

In mathematics, logarithmic Sobolev inequalities are a class of inequalities involving the norm of a function f, its logarithm, and its gradient $\nabla f$ . These inequalities were discovered and named by Leonard Gross, who established them in dimension-independent form,{{harvnb|Gross|1975a}}{{harvnb|Gross|1975b}} in the context of constructive quantum field theory. Similar results were discovered by other mathematicians before and many variations on such inequalities are known.

Gross{{harvnb|Gross|1975a}} proved the inequality:

$\int_{\mathbb{R}^n}\big|f(x)\big|^2 \log\big|f(x)\big| \,d\nu(x) \leq \int_{\mathbb{R}^n}\big|\nabla f(x)\big|^2 \,d\nu(x) +\|f\|_2^2\log \|f\|_2,$

where $\|f\|_2$ is the $L^2(\nu)$ -norm of $f$ , with $\nu$ being standard Gaussian measure on $\mathbb{R}^n.$ Unlike classical Sobolev inequalities, Gross's log-Sobolev inequality does not have any dimension-dependent constant, which makes it applicable in the infinite-dimensional limit.

Entropy functional

Define the entropy functional $\operatorname{Ent}_\mu(f) = \int (f \ln f) d\mu - \int f \ln \left(\int f d\mu\right) d\mu$ This is equal to the (unnormalized) KL divergence by $\operatorname{Ent}_\mu(f) = D_{KL}(f d \mu \| (\int f d\mu) d\mu)$ .

A probability measure $\mu$ on $\mathbb{R}^n$ is said to satisfy the log-Sobolev inequality with constant $C>0$ if for any smooth function f

$\operatorname{Ent}_\mu(f^2) \le C \int_{\mathbb{R}^n} \big|\nabla f(x)\big|^2\,d\mu(x),$

Variants

{{Math theorem

| name = Lemma

| note = {{harv|Tao|2012|loc=Lemma 2.1.16}}

| math_statement = Let $X_1, \dots, X_n$ be random variables that are independent, complex-valued, and bounded. $F: \mathbf{C}^n \rightarrow \mathbf{R}$ be a smooth convex function. Then

$\mathbf{E} F(X) e^{F(X)} \leq\left(\mathbf{E} e^{F(X)}\right)\left(\log \mathbf{E} e^{F(X)}\right)+C \mathbf{E} e^{F(X)}|\nabla F(X)|^2$

for some absolute constant $C$ (independent of $n$ ).

}}

Notes

References

{{Cite book |last=Tao |first=Terence |title=Topics in random matrix theory |date=2012 |publisher=American Mathematical Society |isbn=978-0-8218-7430-1 |series=Graduate studies in mathematics |location=Providence, R.I}}
{{citation|first=Leonard|last=Gross|authorlink=Leonard Gross|year=1975a|title=Logarithmic Sobolev inequalities|journal=American Journal of Mathematics|volume=97|issue=4|pages=1061–1083|doi=10.2307/2373688|jstor=2373688}}
{{citation|first=Leonard|last=Gross|authorlink=Leonard Gross|year=1975b|title=Hypercontractivity and logarithmic Sobolev inequalities for the Clifford-Dirichlet form|journal= Duke Mathematical Journal|volume=42 |issue=3 |pages=383–396|doi=10.1215/S0012-7094-75-04237-4|url=https://projecteuclid.org/euclid.dmj/1077311187}}

Category:Axiomatic quantum field theory

Category:Sobolev spaces

Category:Logarithms