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SUMMARY:On Talagrand's convolution conjecture in Gaussian space - Ronen El
 dan (Berkeley)
DTSTART:20150505T153000Z
DTEND:20150505T163000Z
UID:TALK59125@talks.cam.ac.uk
CONTACT:HoD Secretary\, DPMMS
DESCRIPTION:We consider the Ornstein-Uhlenbeck convolution operator in Gau
 ssian space\, $f\\to P_t[f]$. An easy fact is that every function\, when c
 onvoluted with a small Gaussian noise\, becomes $C_\\infty$ smooth. This r
 aises the question: is there any quantitative way of characterizing how qu
 ickly smoothing occurs under convolution? One natural way to quantify this
  is the so-called hypercontractivity property of the operator $P_t$: for e
 very $t.0$ and $p.1$ there exists $q>p$ such that $P_t$ is a contraction f
 rom  $L_p$ to $L_q$. This property\, which is equivalent to a Log-Sobolev 
 inequality has turned out to be extremely useful in several fields such as
  analysis of PDEs and quantum information theory. However\, this is only m
 eaningful when one has some a priori bound the $L_p$ norm of the initial f
 unction\, for some $p>1$\, and it is not clear if one can say anything abo
 ut singular measures\, for example. IN 1989\, Talagrand conjectured that f
 or any non-negative function $f$ normalized to have integral $1$ over Gaus
 sian space\, the function $P_t[f]$ becomes smooth in the sense that the Ga
 ussian measure of the set ${P_t[f](x) > \\alpha}$ has Gaussian measure $o(
 1/\\alpha)$\, hence $P_t[f]$ satisfies an improved Markov inequality (this
  is dual to a certain isoperimetric-type bound). We prove this conjecture 
 (this is joint work with James Lee). 
LOCATION:MR12\, CMS\, Wilberforce Road\, Cambridge\, CB3 0WB
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