# Analysis and Probability Seminar

The seminar is joint for the division of Analysis and Probability. Unless otherwise indicated, the seminar takes place Tuesdays at 1515-1615 in MVL:14.

18-03-20

Sergey Zuev

Self-decomposable point processes

Self-decomposable (SD) point processes constitute the class of point processes arising as a limit of superpositions of independent point processes. They are subclass of infinitely divisible point processes and contain the class of stable point processes. We characterise SD for the most general scheme involving independent branching operation on points, in particular, thinning. We show that a (regular) point process is SD iff it is representable as a series of branching operations applied to independent realisations of another point process whose scaled distribution correspond to the Levy measure.

The talk is aimed at a general audience familiar with basic probability.

18-04-10
Simone Calogero

Self-gravitating elastic balls

18-04-17
No seminar scheduled due to the workshop "Probabilistic Approaches to Quantum Spin Systems"

18-05-08
Bernt Wennberg
TBA

18-05-15
Boris Vertman (University of Oldenburg)
TBA

18-05-22
Johan Tykesson

TBA

18-05-29

Perla Sousi (University of Cambridge)
TBA

Past Seminars:

18-01-23
The vector-field method for the relativistic transport equation with application to the Einstein-Vlasov system
The vector-field method, developed by Klainerman in the 80s was the key to understand the global existence and asymptotic behaviour of solutions to some nonlinear wave equation. This method is based on the construction of appropriate commutators with the wave equation. Hence, coercive estimates for a solution of a wave equation can be extended to derivatives of this solution, and pointwise weighted estimates can be derived. I will briefly present in this talk this vector-field method, and explain how it extends to relativistic transport equations. Then, after briefly introducing the stability problems in General Relativity, I will explain how this method can be used to prove the stability of the "ground state" of the Einstein equations in the presence of a matter field modeled by a transport equation (Einstein-Vlasov system). This is joint work with David Fajman (Vienna) and Jacques Smulevici (Orsay-ENS) (arXiv:1707.06141).

The first two parts of the talk will be kept as elementary as possible to an audience familiar with basic PDE estimates. The last part of the talk will give a quick and simplified overview on the proof of the stability of Minkowski space.

18-01-30
Hasse Carlsson
Estimates of the renewal measure

Let $\mu$ be a nonlattice probability meassure on the line, and $\nu=\sum_0^\infty \mu^{n*}$ its renewal measure. Blackwell's renewal theorem states that
$$\lim_{x\to+\infty} \nu(x+I)=|I|/\mu_1,$$
where $\mu_1=\int xd\mu(x)$ is the first moment of $\mu$ and $|I|$ the length of the interval $I$.
A lot of papers have studied the rate of this convergence. My talk is yet another attempt in this quest. In particular I am intersted in the case where $\mu$ has finite moments of order $\alpha$ (i.e. $\int|x|^\alpha d\mu(x)<\infty$) when $1>\akpha<2$.
The methods are Fourier analytic.

The talk is aimed at the whole division.

18-02-06
Peter Sjögren
Estimates for some operators associated with the Laplacian with drift in Euclidean space
Let
$v\neq 0$ be a vector in $\mathbb{R}^n$. Consider the Laplacian on $\mathbb{R}^n$ with drift $\Delta_v=\Delta+2v\cdot\Nabla$ and the measure $d\mu(x)=e^{2\langle v,x\rangle}dx$, with respect to which
$\Delta_v$ is self-adjoint. This measure has exponential growth with respect to the Euclidean distance. We study weak type $(1,1)$ and other sharp endpoint estimates for the related Riesz transforms of any order, and also for the vertical and horizontal Littlewood-Paley-Stein functions for the associated heat and Poisson semigroups.
This is joint work with Hong-Quan Li, Shanghai.

The talk is intended for a general audience of analysts.

18-02-20
Przemys
ław Ohrysko

Spectrally reasonable measures

Let us recall that the measure on the circle group is said to have a natural spectrum iff its spectrum is equal to the closure of the image of the Fourier-Stieltjes transform. As the set of all measures does not possess good algebraic properties (it is not closed under addition, in particular) we introduced the concept of spectrally reasonable measures, i. e. a measure perturbing any measure with a natural to spectrum to a measure with a natural spectrum. It appears that the collection of all such measures has a Banach algebra structure. In my talk I will describe a wide class of examples of measures from this class including absolutely continuous ones. Moreover, I will prove that (except trivial cases) no discrete measure is reasonable. The presentation is based on the paper: ‘Spectrally reasonable measures’, St. Petersburg Math. J. 28 (2017), 259-271 written in collaboration with Michał Wojciechowski.

The talk is aimed at everybody familiar with the basics of harmonic analysis and the Banach algebra theory.

18-03-13

Ramiz Reza (Indian Institute of Science, Bangalore)

Curvature inequalities for operators in the Cowen-Douglas class of a planar domain

The talk is aimed at an audience with background and interests in Operator Theory and Complex Analysis.

Published: Mon 12 Dec 2016. Modified: Fri 16 Mar 2018