Boundary value problem and the Ehrhard inequality

Abstract: Let $I, J\subset \mathbb{R}$ be closed intervals, and let $H$ be $C^{3}$ smooth real valued function on $I\times J$ with nonvanishing $H_{x}$ and $H_{y}$. Take any fixed positive numbers $a,b$, and let $d\mu$ be a probability measure with finite moments and absolutely continuous with respect to Lebesgue measure. We show that for the inequality $$ \int_{\mathbb{R}^{n}} \mathrm{ess\,sup}{y \in \mathbb{R}^{n}}\; H\left( f\left(\frac{x-y}{a}\right),g\left(\frac{y}{b}\right)\right)d\mu (x) \geq H\left(\int{\mathbb{R}^{n}}fd\mu, \int_{\mathbb{R}^{n}}gd\mu \right) $$ to hold for all Borel functions $f,g$ with values in $I$ and $J$ correspondingly it is necessary that $$ a^{2}\frac{H_{xx}}{H_{x}^{2}}+(1-a^{2}-b^{2})\frac{H_{xy}}{H_{x}H_{y}}+b^{2}\frac{H_{yy}}{H_{y}^{2}}\geq 0, $$ $|a-b|\leq 1$, $a+b\geq 1$ and $\int_{\mathbb{R}^{n}}xd\mu=0$ if $a+b>1$. Moreover, if $d\mu$ is a Gaussian measure then the necessary condition becomes sufficient. This extends Pr\'ekopa--Leindler and Ehrhard inequalities to an arbitrary function $H(x,y)$. As an immediate application we obtain the new proof of the Ehrhard inequality. In particular, we show that in the class of even probability measures with smooth positive density and finite moments the Gaussian measure is the only one which satisfies the functional form of the Ehrhard inequality on the real line with their own distribution functions.