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Let $f : \mathbb{R}^n \rightarrow [0,\infty)$ be a smooth function and consider $h$ s.t $h(\vec{x}) = f(\vec{x}) + \lambda \Vert \vec{x} \Vert^2$.

  • Does this imply that irrespective of any other property of $f$, the Gibbs' measure of $h$ ($\sim e^{-\beta h}$ for some $\beta >0$) satisfies the log-Sobolev or even just the Poincaré inequality?

I am particularly interested in the case of non-convex $f$.

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  • $\begingroup$ The case of $f$ convex appears to be the strong log-concavity of this survey, though (as detailed on page 6) many authors have called it many things. While you aren't as interested in that case, perhaps this will give you useful terms to search on anyway. $\endgroup$
    – Mark Schultz-Wu
    Commented Oct 6 at 1:25
  • $\begingroup$ Could you be a bit more precise about the smoothness which you have in mind? If f is bounded (or even just of sub-quadratic growth), then the answer is yes; if f has uniformly-bounded second derivatives, then the answer is often yes; if f is just C-infinity, then the answer need not be yes (see e.g. arxiv.org/abs/0810.5435). Happy to comment more as we narrow things down. $\endgroup$
    – πr8
    Commented Oct 6 at 8:05
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    $\begingroup$ I am thinking of $f$ being an unbounded function which is Lipschitz and whose Hessian norm is bounded by a constant. What is the theorem known in this case? $\endgroup$
    – Student
    Commented Oct 6 at 9:28

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Suppose that $\lambda > 0$, $f: \mathbb{R}^n \to \mathbb{R}_+$ is $L$-Lipschitz, and define $$ \mu \left( \mathrm{d} x \right) \propto \exp \left( - \lambda \cdot \| x \|^2 - f\left(x\right) \right). $$ Then, Theorem 1.4 of https://arxiv.org/abs/2404.15205 tells us that there exists a mapping $T : \mathbb{R}^n \to \mathbb{R}^n$ which transports the standard Gaussian measure $\gamma = \mathcal{N} \left( 0, \mathbf{I}_n \right)$ onto $\mu$, and has a Lipschitz constant satisfying $$ \| T \|_\mathrm{Lip} \leq \frac{1}{\sqrt{2\cdot\lambda}} \cdot \exp \left( \frac{L^2}{4 \cdot \lambda} + \frac{2 \cdot L}{\sqrt{2 \cdot \lambda}} \right). $$ As a consequence, it follows that $\mu$ inherits the { LSI, PI, etc. } from $\gamma$ by standard transfer principles.

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  • $\begingroup$ Thanks for this! What is this transfer theorem that you are referring to? $\endgroup$
    – Student
    Commented Oct 6 at 18:23
  • $\begingroup$ There are general results stating that if a probability measure satisfies an energy-entropy inequality of a suitable type (of which LSI and PI are examples), then so does any Lipschitz pushforward of that measure; one can show this quite directly by using the chain rule. I'll try to find a reference. $\endgroup$
    – πr8
    Commented Oct 6 at 22:05
  • $\begingroup$ See e.g. the first two pages of people.math.ethz.ch/~afigalli/papers-pdf/…. $\endgroup$
    – πr8
    Commented Oct 6 at 22:10
  • $\begingroup$ Thanks! It makes sense now. I guess the same argument works with Poincare too. $\endgroup$
    – Student
    Commented Oct 6 at 23:19
  • $\begingroup$ I am curious to know if this Theorem 1.4 you used in the answer is really a 2024 theorem or was this known earlier in some other forms? Is there any old result too that could have led to an affirmative answer to my question? $\endgroup$
    – Student
    Commented Oct 6 at 23:50

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