BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//talks.cam.ac.uk//v3//EN
BEGIN:VTIMEZONE
TZID:Europe/London
BEGIN:DAYLIGHT
TZOFFSETFROM:+0000
TZOFFSETTO:+0100
TZNAME:BST
DTSTART:19700329T010000
RRULE:FREQ=YEARLY;BYMONTH=3;BYDAY=-1SU
END:DAYLIGHT
BEGIN:STANDARD
TZOFFSETFROM:+0100
TZOFFSETTO:+0000
TZNAME:GMT
DTSTART:19701025T020000
RRULE:FREQ=YEARLY;BYMONTH=10;BYDAY=-1SU
END:STANDARD
END:VTIMEZONE
BEGIN:VEVENT
CATEGORIES:Probability
SUMMARY:On Talagrand's convolution conjecture in Gaussian
space - Ronen Eldan (Berkeley)
DTSTART;TZID=Europe/London:20150505T163000
DTEND;TZID=Europe/London:20150505T173000
UID:TALK59125AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/59125
DESCRIPTION:We consider the Ornstein-Uhlenbeck convolution ope
rator in Gaussian space\, $f\\to P_t[f]$. An easy
fact is that every function\, when convoluted with
a small Gaussian noise\, becomes $C_\\infty$ smoo
th. This raises the question: is there any quantit
ative way of characterizing how quickly smoothing
occurs under convolution? One natural way to quant
ify this is the so-called hypercontractivity prope
rty of the operator $P_t$: for every $t.0$ and $p.
1$ there exists $q>p$ such that $P_t$ is a contrac
tion from $L_p$ to $L_q$. This property\, which i
s equivalent to a Log-Sobolev inequality has turne
d out to be extremely useful in several fields suc
h as analysis of PDEs and quantum information theo
ry. However\, this is only meaningful when one has
some a priori bound the $L_p$ norm of the initial
function\, for some $p>1$\, and it is not clear i
f one can say anything about singular measures\, f
or example. IN 1989\, Talagrand conjectured that f
or any non-negative function $f$ normalized to hav
e integral $1$ over Gaussian space\, the function
$P_t[f]$ becomes smooth in the sense that the Gaus
sian measure of the set ${P_t[f](x) > \\alpha}$ ha
s Gaussian measure $o(1/\\alpha)$\, hence $P_t[f]$
satisfies an improved Markov inequality (this is
dual to a certain isoperimetric-type bound). We pr
ove this conjecture (this is joint work with James
Lee).
LOCATION:MR12\, CMS\, Wilberforce Road\, Cambridge\, CB3 0W
B
CONTACT:HoD Secretary\, DPMMS
END:VEVENT
END:VCALENDAR