Boundary value problem and the Ehrhard inequality'

Monday, September 19, 2016 - 3:00pm
Math Sci 312
Paata Ivanishvili (Kent State Univ)

I will present a new proof of the Ehrhard inequality. In fact I will talk about a more general result and the Ehrhard inequality will be consequence of it.  The idea of the method is similar to Brascamp--Lieb's approach to  Prekopa--Leindler inequality via sharp reverse Young's inequality for  convolutions. Indeed, we shall  rewrite  essential supremum as a limit of Lp norms but with very specially chosen test functions and measures. Next  rewriting Lp norm  by duality as a scalar product  the question boils down to  an estimate of double integral of compositions of test functions by the mass of these functions. To verify the last estimate which looks like  Jensen's inequality   we will use a subtle inequality, a ``modified Jensen's inequality'', which in its turn boils down to the fact that a corresponding quadratic form has a definite sign, and this is the main technical part of the method.   If time allows we will show  that in the class of even probability measures  with smooth  strictly positive density  Gaussian measure is the only one which satisfies the functional form of the Ehrhard inequality on the real line with their own distribution function. 

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