On Talagrand's convolution conjecture in Gaussian space

Add to your list(s) Download to your calendar using vCal

• Ronen Eldan (Berkeley)
• Tuesday 05 May 2015, 16:30-17:30
• MR12, CMS, Wilberforce Road, Cambridge, CB3 0WB.

If you have a question about this talk, please contact HoD Secretary, DPMMS.

We consider the Ornstein-Uhlenbeck convolution operator in Gaussian space, $f\to P_t[f]$. An easy fact is that every function, when convoluted with a small Gaussian noise, becomes $C_\infty$ smooth. This raises the question: is there any quantitative way of characterizing how quickly smoothing occurs under convolution? One natural way to quantify this is the so-called hypercontractivity property of the operator $P_t$: for every $t.0$ and $p.1$ there exists $q>p$ such that $P_t$ is a contraction from $L_p$ 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 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 if one can say anything about singular measures, for example. IN 1989 , Talagrand conjectured that for any non-negative function $f$ normalized to have integral $1$ over Gaussian space, the function $P_t[f]$ becomes smooth in the sense that the Gaussian 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).

This talk is part of the Probability series.

This talk is included in these lists:

• All CMS events
• All Talks (aka the CURE list)
• CMS Events
• DPMMS Lists
• DPMMS info aggregator
• DPMMS lists
• Hanchen DaDaDash
• Interested Talks
• MR12, CMS, Wilberforce Road, Cambridge, CB3 0WB
• Probability
• School of Physical Sciences
• Statistical Laboratory info aggregator
• bld31

Note that ex-directory lists are not shown.