world:log-sobolev
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world:log-sobolev [2025/07/09 10:52] rdouc created |
world:log-sobolev [2025/07/09 11:02] (current) rdouc [Concentration inequalities and logarithmic Sobolev inequality] |
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| - | ====== Concentration inequalities and logarithmic Sobolev inequality ====== | + | ====== Logarithmic Sobolev inequality and concentration ====== |
| + | $\newcommand{\Ent}{\mathrm{Ent}}$ | ||
| - | **Theorem.** Let $\mu$ be a probability measure on $\mathbb{R}^n$ satisfying the following logarithmic Sobolev inequality: | + | ====== Theorem ====== |
| + | * Taken from Thm 7.4.1 in the book "Sur les inégalités de Sobolev logarithmiques". | ||
| + | |||
| + | <WRAP center round tip 80%> | ||
| + | Let $\mu$ be a probability measure on $\mathbb{R}^n$ satisfying the following logarithmic Sobolev inequality: | ||
| $$ | $$ | ||
| \forall f \in C_b^\infty(\mathbb{R}^n), \quad \Ent_\mu(f^2) \leq c \, \mathbb{E}_\mu(|\nabla f|^2), | \forall f \in C_b^\infty(\mathbb{R}^n), \quad \Ent_\mu(f^2) \leq c \, \mathbb{E}_\mu(|\nabla f|^2), | ||
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| $$ | $$ | ||
| - | **Proof.** | + | </WRAP> |
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| + | ===== Proof. ===== | ||
| We first assume that $F$ is a smooth and bounded function such that $\|F\|_{\mathrm{Lip}} \leq 1$. Consider the Laplace transform of $F$: | We first assume that $F$ is a smooth and bounded function such that $\|F\|_{\mathrm{Lip}} \leq 1$. Consider the Laplace transform of $F$: | ||
| $$ | $$ | ||
world/log-sobolev.1752051133.txt.gz · Last modified: 2025/07/09 10:52 by rdouc
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