We present a logarithmic Sobolev inequality adapted to a log-concave measure. Assume that $\Phi$ is a symmetric convex function on $\dR$ satisfying $(1+\e)\Phi(x)\leq { x}\Phi'(x)\leq(2-\e)\Phi(x)$ for $x\geq0$ large enough and with $\e\in]0,1/2]$. We prove that the probability measure on $\dR$ $\mu_\Phi(dx)=e^{ -\Phi(x)}/Z_\Phi dx$ satisfies a modified and adapted logarithmic Sobolev inequality : there exist three constant $A,B,D>0$ such that for all smooth $f>0$, \begin{ equation*} \ent{ \mu_\Phi}{ f^2}\leq A\int H_{ \Phi}\PAR{ { \frac{ f'}{ f}}}f^2d\mu_\Phi, \text{ with} H_{ \Phi}(x)= { \begin{ array}{ rl} \Phi^*\PAR{ Bx} &\text{ if }\ABS{ x}\geq D, x^2 &\text{ if}\ABS{ x}\leq D. \end{ array} . \end{ equation*}
Categories: math.PR math.FA