MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am looking for a reference: Does Gaussian measure satisfy Logarithmic Sobolev Inequality (LSI) in $\({\mathbb{R}^d}\)^{\mathbb{Z}^d}$. Thanks.

share|cite|improve this question
Can you clarify what space you're looking at? Do you mean the space of functions $\mathbb{Z}^d \to \mathbb{R}^d$, or equivalently a countably infinite product of copies of $\mathbb{R}^d$? – Mark Meckes Sep 7 '12 at 13:17
If so, and you mean standard Gaussian measure, then this is the same as asking about standard Gaussian measure on $\mathbb{R}^\mathbb{N}$, which should be dealt with in most standard references on LSIs. – Mark Meckes Sep 7 '12 at 13:19
It is the case of standard Gaussian measure on product of countably infinite copies of $\mathbb{R}^d$. Could you please mention any references on this? All I could find is, LSI for standard GM on $\mathbb{R}^d$ but not the product space. – Ahsan Sep 7 '12 at 14:46

As in Mark Meckes's comment, this is equivalent to log-Sobolev for standard Gaussian measure on $\mathbb{R}^\mathbb{N}$, which in turn follows immediately from the finite-dimensional case. In order to show the log-Sobolev inequality $$\int |f|^2 \ln |f| \le \int \|\nabla f\| + \frac{1}{2} \int |f|^2 \ln \int |f|^2$$ it is sufficient to prove it for smooth cylinder functions $f \in L^2(\mathbb{R}^\mathbb{N}, \mu)$, i.e. which depend on only finitely many coordinates. But for $f$ depending on $n$ coordinates, this is precisely the log-Sobolev inequality for standard Gaussian measure on $\mathbb{R}^n$. (Unlike the classical Sobolev inequality, there is no dimension-dependent constant!)

Leonard Gross's original paper introducing the log-Sobolev inequality already mentions the extension to infinite dimensions (though it does not work it out explicitly).

Gross, Leonard. Logarithmic Sobolev Inequalities. American Journal of Mathematics 97 No. 4 (Winter, 1975), pp. 1061-1083.

share|cite|improve this answer
Thanks, Nate. I got busy and didn't get back to this in time. – Mark Meckes Sep 12 '12 at 19:37

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.