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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Degenerate systems of three Brownian particles wit
 h asymmetric collisions: invariant measure of gaps
  and differential properties - Sandro Franceschi (
 Institut polytechnique de Paris)
DTSTART;TZID=Europe/London:20240806T090000
DTEND;TZID=Europe/London:20240806T100000
UID:TALK215659AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/215659
DESCRIPTION:We consider a degenerate system of three Brownian 
 particles which collide asymmetrically. We study t
 he gap process of this system and we focus on its 
 invariant measure. The gap process is an obliquely
  reflected degenerate Brownian motion in a quadran
 t. We fully characterise the cases where the Lapla
 ce transform of the invariant measure is rational\
 , algebraic\, differentially finite or differentia
 lly algebraic. In all these cases\, we determine a
 n explicit formula for the invariant measure in te
 rms of a Theta-like function to which we apply a (
 sometimes fractional) differential operator.\nTo s
 how our main results we start from a kernel functi
 onal equation characterizing the Laplace transform
  of the invariant measure. By an analytic continua
 tion of the Laplace transform and a uniformization
  of the Riemann surface generated by the kernel\, 
 we establish a finite difference equation satisfie
 d by the Laplace transform. Then\, using what we c
 all decoupling functions\, we apply Tutte's invari
 ant approach to solve this equation via conformal 
 gluing functions. Difference Galois theory allows 
 us to find necessary and sufficient conditions for
  the Laplace transform to belong to the hierarchy 
 of functions mentioned above. By taking the invers
 e Laplace transform\, the invariant measure is the
 n computed.\nThis presentation is based on joint w
 ork with T. Ichiba\, I. Karatzas\, and K. Raschel 
 and an upcoming work with T. Dreyfus and J. Flin.
LOCATION:External
CONTACT:
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