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Consider a probability distribution $\pi$ with density $e^{-H(x)}$ on $\mathbb{R}$. Let us say that there is a Poincaré inequality with weight $w$ if for any smooth function $\phi$ satisfying $\int \phi(x) \pi(dx) = 0$ the following inequality holds, $$\int \phi'(x)^2 w^2(x) \pi(dx) \geq \int \phi(x)^2 \pi(dx).$$

Question: given a positive function $h:\mathbb{R} \to (0;+\infty)$, can one compute the optimal weight function $w$ in the sense that $w$ minimizes $\int w^2(x) h^2(x) \pi(dx)$.

Example: if $\pi$ is a Gaussian measure ($i.e$ with $H(x) = \frac{1}{2} x^2$) and $h(x)=1$, playing around with Hermite polynomials, it does not seem very hard to check that $w(x)=\text{Cst}$ is optimal.

In the more general case, it seems like the optimal weight $w_0$ and associated test function $\phi_0$ defined by $\int \phi_0'(x)^2 w_0^2(x) \pi(dx) = \int \phi_0(x)^2 \pi(dx)$ must satisfy $\phi_0'(x)^2 = \lambda h^2(x)$ and $h$ is an eigenfunction of a certain differential operator involving $w_0$. I have not been able to explicitly find $w_0$ by continuing in this direction.

Motivations: given a metric on $\mathbb{R}$ ($i.e$ function $w^2$), one can consider the associated Langevin diffusion that has $\pi$ as invariant distribution. Among all the metric that satisfy certain conditions, which one maximizes the speed of convergence ($i.e$ spectral gap) of the Langevin diffusion towards $\pi$.

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Hi, Alekk do you have any references about the "Spectral Gap" and speed of convergence analysis you are mentionning in your motivations ? Regards – The Bridge Feb 10 '11 at 16:21
the easiest way to understand these concepts might be to first see what is happening in a discrete setting: the book "Markov Chains and Mixing Times" by Levin, Peres, Wilmer is a beautiful introduction to these subjects. – Alekk Feb 10 '11 at 17:38
@Alekk. I'm wondering about your function $h$. What are the typical $h$'s you have in mind? What would be the meaning of $h$ for the Langevin diffusion you mentioned in "Motivations"? Is the $h$ in the paragraph between Example and Motivations the same $h$ of the Question? Could you say a bit more on this? – Hans Mar 14 '12 at 19:09

For various inequalities of this kind see also

S. G. Bobkov and M. Ledoux, Weighted Poincaré-type inequalities for Cauchy and other convex measures. Ann. Probab. 37, No. 2, 403-427 (2009). ArXiv: 0906.1651.

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Standard methods from calculus of variations allow you to compute the optimal weight, but the problem is that, as stated, the optimization is totally unconstrained. You mention that there might be some constraints in the context of the Langevin diffusion. These would be required for you to get non-trivial solutions. For example, any weight function such that $w^{2}(x)\pi(dx)$ becomes a Dirac probability density with non-zero weight only at a singleton set will have zero variance for any function $h(x)$. Your additional constraints will need to ensure that this kind of trivial solution to the optimization doesn't happen. Once you know the constraints, it is just a matter of standard optimization methods (it could be as simple as Lagrange multipliers if your probability measures and the function $h(x)$ are simple enough, as is very often the case in applications such as information theory).

Here is a link to a research paper where a similar idea is put into practice for probability measures that are Cauchy: .

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