Logarithmic Sobolev inequalities

Template:Short description In mathematics, logarithmic Sobolev inequalities are a class of inequalities involving the norm of a function f, its logarithm, and its gradient $\mathrm{\nabla }f$. These inequalities were discovered and named by Leonard Gross, who established them in dimension-independent form, 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 proved the inequality:

$\underset{{ℝ}^n}{\int }|f(x){|}^2\log |f(x)|d\nu (x)\le \underset{{ℝ}^n}{\int }|\mathrm{\nabla }f(x){|}^{2}d\nu (x)+\Vert f{\Vert }_2^2\log \Vert f{\Vert }_2,$

where $\Vert f{\Vert }_2$ is the $L^2(\nu )$-norm of $f$, with $\nu$ being standard Gaussian measure on ${ℝ}^n.$ 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$${\mathrm{Ent}}_{\mu }(f)=\int (f\ln f)d\mu -\int f\ln (\int fd\mu )d\mu$$This is equal to the (unnormalized) KL divergence by ${\mathrm{Ent}}_{\mu }(f)=D_{KL}(fd\mu \Vert (\int fd\mu )d\mu )$.

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

${\mathrm{Ent}}_{\mu }(f^2)\le C\underset{{ℝ}^n}{\int }|\mathrm{\nabla }f(x){|}^{2}d\mu (x),$

References

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