Archive for stochastic processes

Martin Hairer gets Breakthrough Prize (and $3M)

Posted in Books, University life with tags , , , , , , , , , on September 14, 2020 by xi'an

Just heard the news that Fields Medallist Martin Hairer (formerly U of Warwick) got the 2021 Breakthrough Prize in Mathematics for his unification theory of stochastic partial differential equations, which he likens to a form of Taylor expansion in the massive Inventiones paper describing this breakthrough. (Looking at the previous winners of the prize, who also made its selection committee, this represents a break from focussing primarily on algebraic geometry! If not from sticking to male recipients…)

We introduce a new notion of “regularity structure” that provides an algebraic framework allowing to describe functions and/or distributions via a kind of “jet” or local Taylor expansion around each point. The main novel idea is to replace the classical polynomial model which is suitable for describing smooth functions by arbitrary models that are purpose-built for the problem at hand. In particular, this allows to describe the local behaviour not only of functions but also of large classes of distributions. We then build a calculus allowing to perform the various operations (multiplication, composition with smooth functions, integration against singular kernels) necessary to formulate fixed point equations for a very large class of semi-linear PDEs driven by some very singular (typically random) input. This allows, for the first time, to give a mathematically rigorous meaning to many interesting stochastic PDEs arising in physics. The theory comes with convergence results that allow to interpret the solutions obtained in this way as limits of classical solutions to regularised problems, possibly modified by the addition of diverging counterterms. These counterterms arise naturally through the action of a “renormalisation group” which is defined canonically in terms of the regularity structure associated to the given class of PDEs. Our theory also allows to easily recover many existing results on singular stochastic PDEs (KPZ equation, stochastic quantisation equations, Burgers-type equations) and to understand them as particular instances of a unified framework. One surprising insight is that in all of these instances local solutions are actually “smooth” in the sense that they can be approximated locally to arbitrarily high degree as linear combinations of a fixed family of random functions/distributions that play the role of “polynomials” in the theory. As an example of a novel application, we solve the long-standing problem of building a natural Markov process that is symmetric with respect to the (finite volume) measure describing the \Phi^4_ 3 Euclidean quantum field theory. It is natural to conjecture that the Markov process built in this way describes the Glauber dynamic of 3-dimensional ferromagnets near their critical temperature.

from least squares to signal processing and particle filtering

Posted in Books, Kids, Statistics, University life with tags , , , , , , , , on June 6, 2017 by xi'an

Nozer Singpurwalla, Nick. Polson, and Refik Soyer have just arXived a remarkable survey on the history of signal processing, from Gauß, Yule, Kolmogorov and Wiener, to Ragazzini, Shanon, Kálmán [who, I was surprised to learn, died in Gainesville last year!], Gibbs sampling, and the particle filters of the 1990’s.

Sunday morning reading

Posted in Books, Kids, University life with tags , , , , , on June 30, 2016 by xi'an

A very interesting issue of Nature I read this morning while having breakfast. A post-brexit read of a pre-brexit issue. Apart from the several articles arguing against Brexit and its dire consequences on British science [but preaching to the converted for which percentage of the Brexit voters does read Nature?!], a short vignette on the differences between fields for the average time spent for refereeing a paper (maths takes twice as long as social sciences and academics older than 65 half the time of researchers under 36!). A letter calling for action against predatory publishers. And the first maths paper published since I started reading Nature on an almost-regular basis: it studies mean first-passage time for non-Markov random walks. Which are specified as time-homogeneous increments. It is sort of a weird maths paper in that I do not see where the maths novelty stands and why the paper only contains half a dozen formulas… Maybe not a maths paper after all.

métro static

Posted in Kids, Statistics, Travel with tags , , , , , , on July 19, 2015 by xi'an

[In the train shuttle at Birmingham airport, two young guys, maybe back from SPA 2015, discussing signal processing:]

– In Bayesian statistics, they use a different approach to testing hypotheses… You see, they put priors on the different hypotheses…

– But in the end it all boils down to concentration inequalities…

Statistics month in Marseilles (CIRM)

Posted in Books, Kids, Mountains, pictures, Running, Statistics, Travel, University life, Wines with tags , , , , , , , , , , , , , , on June 24, 2015 by xi'an

Calanque de Morgiou, Marseille, July 7, 2010Next February, the fabulous Centre International de Recherche en Mathématiques (CIRM) in Marseilles, France, will hold a Statistics month, with the following programme over five weeks

Each week will see minicourses of a few hours (2-3) and advanced talks, leaving time for interactions and collaborations. (I will give one of those minicourses on Bayesian foundations.) The scientific organisers of the B’ week are Gilles Celeux and Nicolas Chopin.

The CIRM is a wonderful meeting place, in the mountains between Marseilles and Cassis, with many trails to walk and run, and hundreds of fantastic climbing routes in the Calanques at all levels. (In February, the sea is too cold to contemplate swimming. The good side is that it is not too warm to climb and the risk of bush fire is very low!) We stayed there with Jean-Michel Marin a few years ago when preparing Bayesian Essentials. The maths and stats library is well-provided, with permanent access for quiet working sessions. This is the French version of the equally fantastic German Mathematik Forschungsinstitut Oberwolfach. There will be financial support available from the supporting societies and research bodies, at least for young participants and the costs if any are low, for excellent food and excellent lodging. Definitely not a scam conference!