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, 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:

$\int_\big, f(x)\big, ^2 \log\big, f(x)\big, \,d\nu(x) \leq \int_\big, \nabla f(x)\big, ^2 \,d\nu(x) +\, f\, _2^2\log \, f\, _2,$

where $\, f\, _2$ is the $L^2(\nu)$-norm of $f$, with $\nu$ being standard Gaussian measure on $\mathbb^n.$ 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$\operatorname_\mu(f) = \int (f \ln f) d\mu - \int f \ln \left(\int f d\mu\right) d\mu$This is equal to the (unnormalized) KL divergence by $\operatorname_\mu(f) = D_(f d \mu \, (\int f d\mu) d\mu)$.

A probability measure $\mu$ on $\mathbb^n$ is said to satisfy the log-Sobolev inequality with constant $C>0$ if for any smooth function ''f''
$\operatorname_\mu(f^2) \le C \int_ \big, \nabla f(x)\big, ^2\,d\mu(x),$

Variants

Notes

References

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*{{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, url-access=subscription

Axiomatic quantum field theory
Sobolev spaces
Logarithms