grandes-ecoles 2024 Q13

grandes-ecoles · France · mines-ponts-maths1__psi Continuous Probability Distributions and Random Variables Entropy, Information, or Log-Sobolev Functional Analysis

For $t \in \mathbf{R}_{+}$, we set $S(t) = \operatorname{Ent}_{\varphi}\left(P_{t}(f)\right)$. Justify that $S(t)$ is well defined.
Here $f$ is an element of $C^{2}(\mathbf{R})$ with strictly positive values such that the functions $f, f^{\prime}, f^{\prime\prime}$ and $\frac{f^{\prime 2}}{f}$ have slow growth, and $\int_{-\infty}^{+\infty} f(x) \varphi(x) \mathrm{d}x = 1$.

For $t \in \mathbf{R}_{+}$, we set $S(t) = \operatorname{Ent}_{\varphi}\left(P_{t}(f)\right)$. Justify that $S(t)$ is well defined.

Here $f$ is an element of $C^{2}(\mathbf{R})$ with strictly positive values such that the functions $f, f^{\prime}, f^{\prime\prime}$ and $\frac{f^{\prime 2}}{f}$ have slow growth, and $\int_{-\infty}^{+\infty} f(x) \varphi(x) \mathrm{d}x = 1$.

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