Infinite rate mutually catalytic branching driven by alpha-stable Lévy processes

Abstract

The main objective of the present dissertation is to investigate an infinite rate mutually catalytic branching model in one colony, as introduced in [KM10], where the driving Brownian motions are replaced by spectrally positive alpha-stable Lévy processes. To this end, in the first part we examine the exit measure Q of the first quadrant [0,1)^2 of spectrally positive stable processes. Surprisingly, the exit measure of such processes coincides with the one of rho-correlated Brownian motions with the special choice of rho = -cos(pi/alpha) for the correlation parameter. This identity is proved by making use of certain Fredholm-type integral equations for the density functions of Q, which trace back the exit measure of the first quadrant to the exit measure of the upper half-plane. These integral equations are then shown to determine uniquely the density functions of Q. The result can be generalised to the case where the y-axis is rotated by an angle zeta in [0, pi/2). In the second part of the dissertation, we define a Markov process Z which, in analogy to [KM10], can be understood as a mutually catalytic branching process with infinite branching rate (alpha-IMUB). This is done by giving an explicit expression for the transition semigroup in terms of Q. A strong construction as well as a Trotter-type construction is given for that process. We finally show weak convergence of the finite branching rate processes to the alpha-IMUB Z when the branching rate tends to infinity. The right topologisation of the pathspace of càdlàg functions is the Meyer-Zheng pseudo-path topology, which we introduce in Chapter 4. The dissertation also contains an introductory chapter on Lévy processes with an emphasis on stable processes as well as the exit positions of two-dimensional correlated Brownian motions exiting from the wedge and the half-plane.