Poincaré Inequality on Gaussian Measures So I have a working idea on Gaussian-Poincaré Inequality. Namely through the Ornstein-Ullenbeck Generator and Gaussian Integration by parts.
Recently I have stumbled across Sobolev Spaces and have seen there is a Poincaré Inequality defined there as well over an open set $\Omega$ and w.r.t the Lebesgue Measure. This seems more of a Distribution Theory approach which I admit I have just started learning.
My question is is there any connection between the classical poncare inequality defined for functions on a Sobolov Space and the Gaussian-Poincare Inequality.
Any reference or explanation is deeply appreciated.
A simple version of one of the several related results known as the Poincare inequality is the following:
Let $u\in H^1(\Omega)$, where $\Omega \subseteq B(0,R)$ for some $R&gt;0$. Define $\langle u \rangle = \int_\Omega u(x) \, dx$. Then we have
$$ \int_\Omega \left(u(x) - \langle u \rangle \right)^2 \, dx\leq c R^2 \int_\Omega |\nabla u(x)|^2 \, dx$$
A Gaussian Poincare inequality is of a similar form. Let $X$ be a real valued r.v. with a unit normal distribution, and let $f\in C^1(\mathbb{R})$ such that $\mathbb{E}(f(X)^2)$ is finite. Then
$$ \text{Var}(f(X)) = \mathbb{E}\left((f(X) - \mathbb{E}(f))^2 \right) \leq \mathbb{E}(f'(X)^2)$$
You can see how the two results are really the same if you see the expectation operator as just being integration against a different measure.
A simple version of one of the several related results known as the Poincare inequality is the following:
Let $u\in H^1(\Omega)$, where $\Omega \subseteq B(0,R)$ for some $R&gt;0$. Define $\langle u \rangle = \int_\Omega u(x) \, dx$. Then we have
$$ \int_\Omega \left(u(x) - \langle u \rangle \right)^2 \, dx\leq c R^2 \int_\Omega |\nabla u(x)|^2 \, dx$$
A Gaussian Poincare inequality is of a similar form. Let $X$ be a real valued r.v. with a unit normal distribution, and let $f\in C^1(\mathbb{R})$ such that $\mathbb{E}(f(X)^2)$ is finite. Then
$$ \text{Var}(f(X)) = \mathbb{E}\left((f(X) - \mathbb{E}(f))^2 \right) \leq \mathbb{E}(f'(X)^2)$$
You can see how the two results are really the same if you see the expectation operator as just being integration against a different measure.