Mutually catalytic branching at infinite rate [Elektronische Ressource] / Mario Oeler

Mutually Catalytic Branchingat Infinite RateDissertationzur Erlangung des Grades“Doktor der Naturwissenschaften”am Fachbereich “Physik, Mathematik und Informatik”der Johannes Gutenberg-Universita¨tin MainzMario Oelergeboren in Diez an der LahnMainz, den 06.August 2008Datum der mundl¨ ichen Pruf¨ ung: ..............................Dienstag, den 12.November 2008D77 Mainzer DissertationSummaryThe purpose of this doctoral thesis is to prove existence for a mutually catalytic random walkwith infinite branching rate on countably many sites. The process is defined as a weak limit ofan approximating family of processes. An approximating process is constructed by adding jumpsto a deterministic migration on an equidistant time grid. As law of jumps we need to choose theinvariant probability measure of the mutually catalytic random walk with a finite branching ratein the recurrent regime. This model was introduced by Dawson and Perkins (1998) and this thesisrelies heavily on their work. Due to the properties of this invariant distribution, which is in factthe exit distribution of planar Brownian motion from the first quadrant, it is possible to establisha martingale problem for the weak limit of any convergent sequence of approximating processes.