This seminar will be presented in hybrid mode.  The speaker will deliver his talk in person.

Title: Reflected Brownian motion in a cone: a study of the transient case.

Abstract: One of the classic problems in the literature devoted to reflected Brownian motion in a two-dimensional cone is the study of its stationary distribution in the recurrent case. On the other hand, we will focus in this talk on the transient case in order to study the Green’s functions of this process and their asymptotics. This will naturally lead us to consider the Martin boundary of the process which allows us to determine the harmonic functions satisfying oblique Neumann conditions on the edges. For some models, we will illustrate this by studying the probability of escape of the process along an axis or its probability of absorption at the origin. To establish our results, we use analytical methods historically developed in probability and combinatorics to study random walks in the quadrant. We establish functional equations satisfied by the Laplace transforms of Green’s functions and of the probabilities of escape or absorption. Thanks to the theory of boundary value problems (of Riemann and Carleman) it is possible to determine explicit formulas for these transforms involving hypergeometric functions. The saddle point method and transfer lemmas enable us to calculate the asymptotic and establish the Martin boundary.

The talk will be followed by refreshments in the Huxley Common Room at 4pm.