Yan-Xia Ren

Supercritical super-Brownian motion with a general branching mechanism and travelling waves

We oer a probabilistic treatment of the classical problem of existence, uniqueness and asymptotics of monotone solutions to the travelling wave equation associated to the parabolic semi-group equation of a super-Brownian motion with a general branching mechanism. Whilst we are strongly guided by the reasoning in Kyprianou [26] for branching Brownian motion, the current paper oers a number of new insights. Our analysis incorporates the role of Seneta-Heyde norming which, in the current setting, draws on classical work of Grey [20]. We give a pathwise explanation of Evans’ immortal particle picture (the spine decomposition) which uses the Dynkin-Kuznetsov N-measure as a key ingredient. Moreover, in the spirit of Neveu’s stopping lines we make repeated use of Dynkin’s exit measures. Additional complications arise from the general nature of the branching mechanism. As a consequence of the analysis we also oer an exact X(logX) 2 moment dichotomy for the almost sure convergence of the socalled derivative martingale at its critical parameter to a non-trivial limit. This diers to the case of branching Brownian motion, [26], and branching random walk, [2], where a moment ‘gap’ appears in the necessary and sucient conditions. Our probabilistic treatment allows us to replicate known existence, uniqueness and asymptotic results for the travelling wave equation, which is related to a super-Brownian motion.

The Backbone Decomposition for Spatially Dependent Supercritical Superprocesses

Consider any supercritical Galton-Watson process which may become extinct with positive probability. It is a well-understood and intuitively obvious phenomenon that, on the survival set, the process may be pathwise decomposed into a stochastically ‘thinner’ Galton-Watson process, which almost surely survives and which is decorated with immigrants, at every time step, initiating independent copies of the original Galton-Watson process conditioned to become extinct. The thinner process is known as the backbone and characterizes the genealogical lines of descent of prolific individuals in the original process. Here, prolific means individuals who have at least one descendant in every subsequent generation to their ownn.

Interior singularity problem of some nonlinear elliptic equations

Let L be a uniformly elliptic operator in Rd. We investigate positive solutions to the interior singularity problem of the nonlinear equation Lu=u[alpha],1

Séminaire de Probabilités XLVI

This volume provides a broad insight on current, high level researches in probability theory.

Multitype branching Brownian motion and traveling waves

In this article we study the parabolic system of equations which is closely related to a multitype branching Brownian motion. Particular attention is paid to the monotone traveling wave solutions of this system. Provided with some moment conditions, we show the existence, uniqueness, and asymptotic behaviors of such waves with speed greater than or equal to a critical value c̲ and nonexistence of such waves with speed smaller than c̲.