Confined random processes and Galois theory of difference equations

Vortrag von Prof. Dr. Kilian Raschel

Sprecher eingeladen von: Prof. Dr. Jean Bertoin

Datum: 20.11.19  Zeit: 17.15 - 18.15  Raum: ETH HG G 43

Some models of random walks or Brownian motions in cones can be studied via functional equations that satisfy their Laplace transforms or generating functions. This approach applies to various situations, allowing to study first passage times, extinction probabilities, numbers of paths, stationary probabilities for reflected Brownian motion, etc.

It is rather easy to reformulate these functional equations in terms of q-difference equations, for example f(q*s) - f(s) = g(s), where f is the unknown function (typically the generating function), while g and q are known and depend on the model. Tools from the theory of difference equations are then perfectly adapted, in particular to characterize the algebraic nature of the solution, or even to compute it.

In this talk we will present several examples: we will begin by the enumeration of quadrant walks, for which recent works by Dreyfus, Hardouin, Roques and Singer characterize the differential transcendence of the generating functions. We will also study planar reflected Brownian motion and present a joint work with Bousquet-Mélou, Elvey Price, Franceschi and Hardouin, giving a complete characterization of the Laplace transforms of the stationary distributions. We will conclude by showing some open problems.