Let $m$ be a measure. Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function that admits a variance relative to $m$. Show that $fm$ is integrable. As a consequence, the real $$\operatorname { Var } _ { m } ( f ) = \int f ( x ) ^ { 2 } m ( x ) d x - \left( \int f ( x ) m ( x ) d x \right) ^ { 2 }$$ is well defined. Show that $\operatorname { Var } _ { m } ( f ) \geqslant 0$.
Let $m$ be a measure. Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function that admits an entropy relative to $m$. We consider the function $h : [ 0 , + \infty [ \rightarrow \mathbb { R }$ defined by $h ( 0 ) = 0$ and for $x > 0$, $h ( x ) = x \ln ( x )$. 2a. Show that $f ^ { 2 } m$ is integrable. As a consequence, the real $$\operatorname { Ent } _ { m } ( f ) = \int h \left( f ( x ) ^ { 2 } \right) m ( x ) d x - h \left( \int f ( x ) ^ { 2 } m ( x ) d x \right)$$ is well defined. 2b. Let $a > 0$. Show that $$\forall x \geqslant 0 , \quad h ( x ) \geqslant ( x - a ) h ^ { \prime } ( a ) + h ( a ) ,$$ with strict inequality if $x \neq a$. 2c. Show that $\operatorname { Ent } _ { m } ( f ) \geqslant 0$. You may use the previous question with $a = \int f ( x ) ^ { 2 } m ( x ) d x$. 2d. We assume here that for all $x \in \mathbb { R } , m ( x ) > 0$. Characterize the functions $f$ such that $\operatorname { Ent } _ { m } ( f ) = 0$.
We denote $L$ the operator that associates to a function $f : \mathbb { R } \rightarrow \mathbb { R }$ of class $\mathscr { C } ^ { 2 }$, the function $Lf$ defined by $$\forall x \in \mathbb { R } , \quad L f ( x ) = \frac { 1 } { 2 } f ^ { \prime \prime } ( x ) - x f ^ { \prime } ( x )$$ We recall that the measure $\mu$ is defined by $\mu ( x ) = \frac { 1 } { \sqrt { \pi } } e ^ { - x ^ { 2 } }$. 3a. Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be of class $\mathscr { C } ^ { 2 }$. Show that $L f = \frac { 1 } { 2 \mu } \left( \mu f ^ { \prime } \right) ^ { \prime }$. 3b. Let $h _ { 1 } , h _ { 2 }$ be two functions in $\mathscr { C } _ { b } ^ { 2 }$. Show that $$\int h _ { 1 } ( x ) \left( L h _ { 2 } \right) ( x ) \mu ( x ) d x = - \frac { 1 } { 2 } \int h _ { 1 } ^ { \prime } ( x ) h _ { 2 } ^ { \prime } ( x ) \mu ( x ) d x$$ after having justified the existence of each term of the formula.
We consider a function $f \in \mathscr { C } _ { b } ^ { 0 }$. We define for $( t , x ) \in \mathbb { R } ^ { 2 }$ $$\Phi _ { f } ( t , x ) = \int f ( x \cos t + y \sin t ) \mu ( y ) d y$$ where $\mu ( x ) = \frac { 1 } { \sqrt { \pi } } e ^ { - x ^ { 2 } }$. Show that the function $\Phi _ { f } : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R }$ is well defined and continuous.
We consider a function $f \in \mathscr { C } _ { b } ^ { 0 }$. We define for $( t , x ) \in \mathbb { R } ^ { 2 }$ $$\Phi _ { f } ( t , x ) = \int f ( x \cos t + y \sin t ) \mu ( y ) d y$$ where $\mu ( x ) = \frac { 1 } { \sqrt { \pi } } e ^ { - x ^ { 2 } }$, and $Lf(x) = \frac{1}{2} f''(x) - x f'(x)$. We assume that $f \in \mathscr { C } _ { b } ^ { 2 }$. 5a. Show that, on $\mathbb { R } ^ { 2 } , \Phi _ { f }$ is of class $\mathscr { C } ^ { 1 }$ and $\partial _ { x x } \Phi _ { f }$ is well defined, continuous and bounded. 5b. Let $( t , x ) \in \mathbb { R } ^ { 2 }$. Find a relation between $\partial _ { x } \Phi _ { f } ( t , x )$ and $\Phi _ { f ^ { \prime } } ( t , x )$. 5c. Show that for all $( t , x ) \in \mathbb { R } ^ { 2 }$, we have $\partial _ { t } \Phi _ { f } ( t , x ) \cos t = L \Phi _ { f } ( t , x ) \sin t$. 5d. Show that for all $t \in \mathbb { R }$, we have $\int \Phi _ { f } ( t , x ) \mu ( x ) d x = \int f ( x ) \mu ( x ) d x$.
Let $f : \mathbb { R } \rightarrow \mathbb { R } _ { + }$ be a positive function in $\mathscr { C } _ { b } ^ { 0 }$. We define for $t \in \mathbb { R }$ $$J ( t ) = \int h \left( \Phi _ { f } ( t , x ) \right) \mu ( x ) d x$$ where $\Phi _ { f } ( t , x ) = \int f ( x \cos t + y \sin t ) \mu ( y ) d y$, $\mu ( x ) = \frac { 1 } { \sqrt { \pi } } e ^ { - x ^ { 2 } }$, and $h(x) = x\ln(x)$ for $x > 0$, $h(0) = 0$. Show that $J : \mathbb { R } \rightarrow \mathbb { R }$ is continuous, and calculate $J ( 0 )$ and $J \left( \frac { \pi } { 2 } \right)$.
Let $f : \mathbb { R } \rightarrow \mathbb { R } _ { + }$ be a positive function in $\mathscr { C } _ { b } ^ { 0 }$. We define for $t \in \mathbb { R }$ $$J ( t ) = \int h \left( \Phi _ { f } ( t , x ) \right) \mu ( x ) d x$$ where $\Phi _ { f } ( t , x ) = \int f ( x \cos t + y \sin t ) \mu ( y ) d y$, $\mu ( x ) = \frac { 1 } { \sqrt { \pi } } e ^ { - x ^ { 2 } }$, and $h(x) = x\ln(x)$ for $x > 0$, $h(0) = 0$. We assume throughout this question that $f \in \mathscr { C } _ { b } ^ { 2 }$ and that there exists $\delta > 0$ such that $$\forall x \in \mathbb { R } , \quad f ( x ) \geqslant \delta .$$ We denote $g = \left( f ^ { \prime } \right) ^ { 2 } / f$. 7a. Show that $J$ is then of class $\mathscr { C } ^ { 1 }$ on $\mathbb { R }$ and that $$\forall t \in \mathbb { R } , \quad J ^ { \prime } ( t ) \cos t = - \frac { \sin t } { 2 } \int \frac { \left( \partial _ { x } \Phi _ { f } ( t , x ) \right) ^ { 2 } } { \Phi _ { f } ( t , x ) } \mu ( x ) d x$$ 7b. Let $( t , x ) \in \mathbb { R } ^ { 2 }$. Show that $$\Phi _ { f ^ { \prime } } ( t , x ) ^ { 2 } \leqslant \Phi _ { f } ( t , x ) \Phi _ { g } ( t , x )$$ 7c. Conclude that $$\int h ( f ( x ) ) \mu ( x ) d x - h \left( \int f ( y ) \mu ( y ) d y \right) \leqslant \frac { 1 } { 4 } \int g ( x ) \mu ( x ) d x$$