Abstract

We consider a degenerate system of three Brownian particles which 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, 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 main results we start from a kernel functional equation characterizing the Laplace transform of the invariant measure. By an analytic continuation of the Laplace transform and a uniformization of the Riemann surface generated by the kernel, we establish a finite difference equation satisfied by the Laplace transform. Then, using what we call decoupling functions, we apply Tutte’s invariant 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 inverse Laplace transform, the invariant measure is then computed. 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.