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

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 absence of spectral gaps

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.

 

New team member: Frank Rösler

On 1 October 2018 we were joined by a new team member, Dr Frank Rösler. Frank completed his PhD at Durham University and worked as a Research Assistant in Freiburg (Germany).

He is interested in the spectral theory of non-selfadjoint operators and other operator-theoretic questions in PDE theory. His past projects involved pseudospectra of non-normal Schrödinger Operators and more general resolvent norm estimates of partial differential operators. More recently, he studied problems in Asymptotic Analysis and Homogenisation from an operator-theoretic perspective.

Welcome Frank!

Postdoc position advertised

We are in the process of hiring another postdoc to join our team (see current members). We are looking for someone with research interests in analysis, and, more precisely, someone who is interested in how to approximate infinite dimensional objects in finite dimensions. A typical example is approximating operators in Hilbert space. If you think this sounds interesting please apply. Of course, you will also be free to continue your own research.

The advert is here.

This will involve joint work with our colleagues Marco Marletta (Cardiff) and Anders Hansen (Cambridge).

Marie Skłodowska-Curie Fellowship Success

Jonathan Ben-Artzi and Junyong Zhang have been awarded a Marie Skłodowska-Curie Fellowship which will commence on 1 July 2018 for a period of two years. Their project, entitled “Geometric Analysis of Dilute Plasmas” (GRANDPA), will focus on studying regularity theory and long-time behavior of plasmas governed by the Vlasov-Maxwell system. The abstract reads:

“The ultimate goal of this Fellowship is to understand the long time behaviour of plasmas governed by the relativistic Vlasov- Maxwell system (RVM). The main difficulty is the hyperbolic nature of Maxwell’s equations (the electromagnetic fields propagate at the speed of light): particles that travel close to the speed of light nearly interact with their own fields. It is not currently known whether particles can be accelerated to such speeds, and, if so, whether this necessarily leads to development of singularities. This is a major open problem.”

 

The combined expertise of Jonathan and Junyong in kinetic theory and in dispersive equations played a central role in the success of this application. The total value of the award is €195,455.