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SUMMARY:Degenerate systems of three Brownian particles with asymmetric col
 lisions: invariant measure of gaps and differential properties - Sandro Fr
 anceschi (Institut polytechnique de Paris)
DTSTART:20240806T080000Z
DTEND:20240806T090000Z
UID:TALK215659@talks.cam.ac.uk
DESCRIPTION:We consider a degenerate system of three Brownian particles wh
 ich collide asymmetrically. We study the gap process of this system and we
  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 rational\, algebr
 aic\, differentially finite or differentially algebraic. In all these case
 s\, we determine an explicit formula for the invariant measure in terms of
  a Theta-like function to which we apply a (sometimes fractional) differen
 tial operator.\nTo show our main results we start from a kernel functional
  equation characterizing the Laplace transform of the invariant measure. B
 y an analytic continuation of the Laplace transform and a uniformization o
 f the Riemann surface generated by the kernel\, we establish a finite diff
 erence equation satisfied by the Laplace transform. Then\, using what we c
 all decoupling functions\, we apply Tutte's invariant approach to solve th
 is equation via conformal gluing functions. Difference Galois theory allow
 s us to find necessary and sufficient conditions for the Laplace transform
  to belong to the hierarchy of functions mentioned above. By taking the in
 verse Laplace transform\, the invariant measure is then computed.\nThis pr
 esentation is based on joint work with T. Ichiba\, I. Karatzas\, and K. Ra
 schel and an upcoming work with T. Dreyfus and J. Flin.
LOCATION:External
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