COOKIES: By using this website you agree that we can place Google Analytics Cookies on your device for performance monitoring. |

University of Cambridge > Talks.cam > Probability > On Talagrand's convolution conjecture in Gaussian space

## On Talagrand's convolution conjecture in Gaussian spaceAdd 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
- MR12, CMS, Wilberforce Road, Cambridge, CB3 0WB
- Probability
- School of Physical Sciences
- Statistical Laboratory info aggregator
Note that ex-directory lists are not shown. |
## Other listsShort Course: Higher order regularisation in imaging Cambridge Radiology Forum Philosophy and Natural Philosophy in the Early Modern Period## Other talksEpigenetics: One Genome, Multiple Phenotypes Aspects of adaptive Galerkin FE for stochastic direct and inverse problems Genes against beans: favism, malaria and nationalism in the Middle East The DNA oxygenase TET1 in mammalian embryonic development and epigenetic reprogramming Polish Britain: Multilingualism and Diaspora Community Poland, Europe, Freedom: A Personal Reflection on the Last 40 Years |