Speaker: Sandro Franceschi (Télécom SudParis)

Title: Degenerate systems of three Brownian particles with asymmetric collisions

Abstract:

We consider a degenerate system of three particles which collide asymmetrically. The system is degenerate because the particle in the middle is Brownian and the others are ballistic. We study this system's gap process and focus on its invariant measure. The gap process is an obliquely reflected degenerate Brownian motion in a quadrant. We fully characterise the cases where the Laplace transform of the invariant measure is simple, that is rational, algebraic, differentially finite or differentially algebraic. In all these cases, we determine an explicit formula for the invariant measure in terms of a Theta-like function to which we apply a (sometimes fractional) differential operator.

To show our results, we start with a functional equation that characterises the Laplace transform of the invariant measure, derive a finite difference equation, and use Tutte’s invariant approach and some Difference Galois theory.

This presentation is based on joint work with T. Ichiba, I. Karatzas, and K. Raschel and an upcoming work with T. Dreyfus and J. Flin.

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Note: Back-to-back seminars with different times.​​​​​​​