Anton Wakolbinger

Growing conditioned trees

For a Markovian branching particle system in d a Palm type distribution on the genealogical trees up to a time horizon t is computed, which generically (i.e. if there are almost surely no multiplicities in the particle positions at time t) can be viewed as a conditional distribution on the trees given that the particle system at time t populates a certain site. The result is obtained in two different ways: by conditioning on the first branching and by means of Kallenbergs method of backward trees.

Occupation Time Fluctuations in Branching Systems

AbstractWe consider particle systems in locally compact Abelian groups with particles moving according to a process with symmetric stationary independent increments and undergoing one and two levels of critical branching. We obtain long time fluctuation limits for the occupation time process of the one- and two-level systems. We give complete results for the case of finite variance branching, where the fluctuation limits are Gaussian random fields, and partial results for an example of infinite variance branching, where the fluctuation limits are stable random fields. The asymptotics of the occupation time fluctuations are determined by the Green potential operator G of the individual particle motion and its powers G2,G3, and by the growth as t→∞ of the operator

Ergodic behavior of locally regulated branching populations

G_t = \int_0^t {T_s } ds

A simplified variational characterization of Schrödinger processes

and its powers, where Tt is the semigroup of the motion. The results are illustrated with two examples of motions: the symmetric α-stable Lévy process in

On Free Energy, Stochastic Control, and Schrödinger Processes

\mathbb{R}^d (0 < \alpha \leqslant 2)

Schrödinger processes and large deviations

, and the so called c-hierarchical random walk in the hierarchical group of order N (0<c<N). We show that the two motions have analogous asymptotics of Gt and its powers that depend on an order parameter γ for their transience/recurrence behavior. This parameter is γ=d/α−1 for the α-stable motion, and γ=log c/log(N/c) for the c-hierarchical random walk. As a consequence of these analogies, the asymptotics of the occupation time fluctuations of the corresponding branching particle systems are also analogous. In the case of the c-hierarchical random walk, however, the growth of Gt and its powers is modulated by oscillations on a logarithmic time scale.