# New paper: Uniform convergence in von Neumann’s ergodic theorem in the absence of a spectral gap

*** Update: this is now published in Ergod. Th. & Dynam. Sys. ***

Jonathan Ben-Artzi and Baptiste Morisse recently submitted a paper entitled Uniform convergence in von Neumann’s ergodic theorem in absence of a spectral gap.

Von Neumann’s ergodic theorem states that “time” averages converge to “spatial” avergaes: given a one-parameter family of unitary maps $U_t:\mathcal{H}\to\mathcal{H},\,t\in\mathbb{R},$ the average $\frac{1}{2T}\int_{-T}^TU_tf\,\mathrm{d}t$ converges to the projection of $f$ onto the space of functions invariant under $U_t$ as $T\to+\infty$.

Generally there is no rate. However, if the generator of $U_t$ has a spectral gap, the rate is $T^{-1}$. In the present paper, it is shown that even in the absence of a spectral gap one can obtain a rate, albeit on a subspace of $\mathcal{H}$, and with a rate worse than $T^{-1}$. This is done by obtaining a suitable estimate for the density of the spectrum near zero (low frequencies).

# New paper: Weak Poincaré inequalities in the absence of spectral gaps

*** Update: this paper is now published in Ann. Henri Poincaré ***

Jonathan Ben-Artzi recently uploaded a new paper entitled Weak Poincaré inequalities in absence of spectral gaps, co-authored with Amit Einav.

For Markov semigroups it is well-known that the following are equivalent:

• The generator has a spectral gap,
• The generator satisfies a Poincaré inequality,
• Solutions decay exponentially

In this paper, they study semigroups which lack a spectral gap (such as the heat semigroup in $\mathbb{R}^d$) and try to see how much of the above theorem remains true. They prove that an estimate on the density of the spectrum near 0 leads to a weak Poincaré inequality, which in turn leads to an algebraic decay rate.

This is applied to the heat semigroup, where the optimal decay rate $t^{-d/2}$ is recovered. In this case, the weak Poincaré inequality is no more than the Nash inequality. This is done for the fractional Laplacian as well, with similar results.

# Research journey to Lyon

Thanks to his LMS Scheme 3 grant worth £1200, Baptiste enjoyed a two weeks stay in Lyon from November 10 to 25. Baptiste worked with his collaborator Francesco Fanelli on a new project that aims to explore the question of the inviscid limit for some hydrodynamical equations – as the Euler equation with variable density – in the wake of the important work of Jacob Bedrossian and Nader Masmoudi on Euler equations. This project lies mathematically at the intersection of Baptiste’s line of research on Gevrey regularity and Francesco’s knowledge on hydrodynamical systems.

Baptiste was also invited to the Analysis Seminar of Besançon and gave a talk on weakly hyperbolic systems. The audience was very interested by this subject and raised many interesting questions. It may be the sign that Baptiste will go back to Besançon in the near future!

# Talks & Travels

In his second month here at Cardiff Frank Rösler has been invited to the University of Cambridge and Imperial College London to give seminar talks. These visits provided the opportunity to discuss his research with a wider audience and make progress on open questions from his recent research project on spectral approximation.

On 14th December he will attend the next meeting of the South Wales Analysis & Probability Seminar in Swansea.

# An Analyst, a Geometer and a Probabilist Walk Into a Bar

Last month (25-29 June 2018) we hosted a successful workshop in Cardiff, entitled An Analyst, a Geometer and a Probabilist Walk Into a Bar. There were about 40 participants from Austria, Canada, France, Israel, Italy, Portugal, Spain, Switzerland, the UK and USA. These included many students and early career researchers. Here is the workshop photo:

# Talks & travels

The past eight months have been quite busy for Baptiste Morisse, giving talks in several Analysis Seminar in Lyon, Imperial College, Queen’s Mary… Baptiste also participated in the second session of the UK Network on Hyperbolic Equations, hosted at Loughborough University. These talks were the occasion to meet many interesting people and to build new relations between our group and other researchers in related fields.

To end this series of talks and journeys, Baptiste is now in a two weeks journey in the USA. First to join the conference at Brown for the 80th birthday of Walter Strauss. Then to travel to Golden, Colorado to work with Steve Pankavich, one of the main collaborators of our team. Hope this week will be fruitful!

# New paper: Arbitrarily Large Solutions of the Vlasov-Poisson System

Jonathan Ben-Artzi recently uploaded a new paper entitled Arbitrarily large solutions of the Vlasov-Poisson system, co-authored with Stephen Pankavich and Simone Calogero.

The Vlasov-Poisson system, which models the statistical behavior of many-particle systems, is known to have global-in-time classical solutions (in three dimensions). However, the underlying particle systems (of attractive or repulsive particles) may have singularities appearing in finite time. For instance, attractive particles (stars) can collapse to a single point in finite time. It is therefore interesting to ask how close to a singularity can the Vlasov-Poisson system get?

This has recently been done by Rein & Taegert in the attractive case, however the repulsive case remained open. The main result states that for any constants $C_1,C_2>0$ there exists initial data with density whose $L^\infty$ norm is initially bounded by $C_1$ but that at some later time $T>0$ is greater than $C_2$. The main tool is obtaining a priori estimates for particle trajectories and choosing initial data carefully. This data is chosen to be supported on a spherical shell about the origin, with initial velocities pointing inwards.

# Talks in Princeton and Montréal

Last month Jonathan Ben-Artzi gave talks in the analysis seminar at Princeton University and at the Montréal Analysis Seminar hosted at McGill University. Following these successful talks, we expect a busy time traveling elsewhere and hosting colleagues in Cardiff.