## LSI for Gaussian measure in $({\mathbb{R}^d})^{\mathbb{Z}^d}$

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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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 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 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 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!)