Lectures

ABSTRACT

The propagation of a (quantum) wave in a random medium and the derivation of the radiative transfer equation or the linear Boltzmann equation, is a fundamental problem which has received several types of mathematical answers in the last decades. Although those results are in accordance with the physical models and sometimes suffice for additional modelling work, they are still puzzling : either their proof technically diverge from the obvious heuristic arguments or they seem incomplete from a mathematical point of view. Several models, eg. translation invariant gaussian or poissonian potentials in the weak or low density limit, can be formulated in the framework of bosonic QFT where the number of “particles” is a translation of the probalistic “chaos decomposition”. The different frameworks allow a wide variety of techniques, which make this problem extremely rich from a mathematical point of view. A key point, although it is not the only one, is about accurate number estimates. By combining the endpoint Strichartz estimates by Keel and Tao with an adaptation of the proof of Cauchy-Kowalevski theorem, we obtained with Sébastien Breteaux in a work in progress, such accurate number estimates. They can be used to get refined a priori pieces of information for a general class of initial data and essentially reduce the problem to finite dimensional semiclassical and microlocal analysis issues.

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