21-22

Weak solutions to the homogeneous Boltzmann equation with increasing energy have been constructed by Lu and Wennberg. We consider an underlying microscopic stochastic model with binary collisions and show that these solutions are atypical.

More precisely, we prove that the probability of observing these paths is exponentially small in the number of particles and compute the exponential rate. 

The Fröhlich polaron describes a quantum particle in a polar crystal. By the interaction with the field modes of the crystal, the particle appears heavier than it is, that is referred to as effective mass. An old conjecture of Landau and Pekar is that this effective mass should scale like $\alpha^4$ in the coupling parameter $\alpha$ that measures the strength of the particle/field interaction. It has been proved recently by Lieb and Seiringer that the effective mass diverges as $\alpha \to \infty$, but there were no quantitative lower bounds. 

I will present new results obtained jointly with Steffen Polzer (Geneva) that go 10% of the way towards the Landau/Pekar conjecture: we show that the effective mass is bounded below by a constant time $\alpha^{2/5}$. The method uses a recent representation of the Polaron path measure due to Mukherjee and Varadhan, describing the polaron as a statistical mechanics system of interacting intervals on the real line. 

After introducing some experimental evidence for the occurrence of a percolative metal-to-superconductor transition in superconductors with disorder at the nanoscale, I will discuss the theory of transport in random media, and its description within the effective medium theory. Finally, I will compare the results obtained within the effective medium theory with those obtained by means of the numerical solution of random resistor networks, in the absence and in the presence of spatial correlations within the superconducting cluster.

Information. This seminar is part of the Stochastic Modelling in l'Aquila (SMAQ) seminar series, jointly organised with the University of L'Aquila, and one of the Conferences of the Centro Linceo (link). 

A two-layer quasi-geostrophic (2LQG) model for geophysical flows is studied, with the upper layer being perturbed by additive noise. This model is popular in the geosciences, for instance to study the effects of a stochastic wind forcing on the ocean. A rigorous mathematical analysis however meets with the challenge that the noise configuration is spatially degenerate as the stochastic forcing acts only on the top layer. We will discuss the problem of unique ergodicity and exponential convergence of transition probabilities as well as response theory for stochastic partial differential equations like the 2LQG model.