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Seminarier i Matematisk statistik

För mer information, kontakta Torkel Erhardsson.

 

Aktuella seminarier

Tisdag 5 juni 2018, Vlad Bally, Université Paris-Est Marne-la-Vallée, Frankrike

Seminariet är ett samarrangemang med Matematiska kollokviet.

Talare: Vlad Bally, Université Paris-Est Marne-la-Vallée, Frankrike

Titel: Asymptotic integration by parts formula and regularity of probability laws

Tid och plats: Tisdag 5 juni 2018, Hopningspunkten, 15.15-16.15

Sammanfattning: We consider a sequence of random variables $F_{n}\sim p_{n}(x)dx$ which converge to a random variable $F.$ If we know that $p_{n}\rightarrow p$ in some sweated sense, then we obtain $F\sim p(x)dx.$ But in many interesting situations $p_{n}$ blows up as $n\rightarrow \infty .$ Our aim is to give a criterion which says that, if there is a "good equilibrium" between $\left\Vert F-F_{n}\right\Vert _{1}\rightarrow 0$ and $\left\Vert p_{n}\right\Vert \uparrow \infty $ then we are still able to obtain the absolute continuity of the law of $F$ and to study the regularity of the density $p.$ Moreover we get some upper bounds for $p.$ The blow up of $p_{n}$ is characterized in terms of integration by parts formulae.

We give two examples. The first one is about diffusion processes with Hölder coefficients. The second one concerns the solution $f_{t}(dv)$ of the two dimensional homogeneous Boltzmann equation. We prove that, under some conditions on the parameters of the equation, we have $f_{t}(dv)=f_{t}(v)dv.$ The initial distribution $f_{0}(dv)$ is a general measure (except a Dirac mass) so our result says that a regularization effect is at work; moreover, if the initial distribution has exponential moments $\int e^{\left\vert v\right\vert ^{\lambda }}f_{0}(dv)<\infty ,$ then we prove that $f_{t}(v)\leq Ct^{-\eta }e^{-\left\vert v\right\vert ^{\lambda ^{\prime }}}$ for every $\lambda ^{\prime }<\lambda .$ So we have exponential upper bounds in space and at most polynomial blow up in time.


Thu Apr 05 11:11:00 CEST 2018



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