grandes-ecoles 2024 Q17

grandes-ecoles · France · mines-ponts-maths1__psi Continuous Probability Distributions and Random Variables Entropy, Information, or Log-Sobolev Functional Analysis
By admitting that the result of question 7 is valid for the functions $P_{t}(f)$ and $1 + \ln\left(P_{t}(f)\right)$, show that
$$\forall t \in \mathbf{R}_{+}^{*}, \quad -S^{\prime}(t) = \mathrm{e}^{-2t} \int_{-\infty}^{+\infty} \frac{P_{t}\left(f^{\prime}\right)(x)^{2}}{P_{t}(f)(x)} \varphi(x) \mathrm{d}x$$ 
By admitting that the result of question 7 is valid for the functions $P_{t}(f)$ and $1 + \ln\left(P_{t}(f)\right)$, show that

$$\forall t \in \mathbf{R}_{+}^{*}, \quad -S^{\prime}(t) = \mathrm{e}^{-2t} \int_{-\infty}^{+\infty} \frac{P_{t}\left(f^{\prime}\right)(x)^{2}}{P_{t}(f)(x)} \varphi(x) \mathrm{d}x$$
Paper Questions
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