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.


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.