I am looking for a reference: Does Gaussian measure satisfy Logarithmic Sobolev Inequality (LSI) in $\({\mathbb{R}^d}\)^{\mathbb{Z}^d}$. Thanks.
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$\begingroup$ 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 $\mathbb{R}^d$? $\endgroup$– Mark MeckesSep 7, 2012 at 13:17
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2$\begingroup$ 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. $\endgroup$– Mark MeckesSep 7, 2012 at 13:19
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1$\begingroup$ 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. $\endgroup$– AhsanSep 7, 2012 at 14:46
1 Answer
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.
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$\begingroup$ Thanks, Nate. I got busy and didn't get back to this in time. $\endgroup$ Sep 12, 2012 at 19:37