Sylvie Roelly

Persistence of critical multitype particle and measure branching processes

SummaryWe consider a class of systems of particles ofk types inRd undergoing spatial diffusion and critical multitype branching, where the diffusions, the particle lifetimes and the branching laws depend on the types. We prove persistence criteria for such systems and for their corresponding high density limits known as multitype Dawson-Watanabe processes. The main tool is a representation of the Palm distributions for a general class of inhomogeneous critical branching particle systems, constructed by means of a “backward tree”.

Invariance principle for martingale-difference random fields

A convergence criterium to the multi-parameter Wiener process is proved. Then, it is used to establish that a martingale-difference random field on the lattice satisfies an invariance principle.

Some properties of the multitype measure branching process

Qualitative properties of the multitype measure branching process and its occupation time process are investigated, including martingale properties, Hausdorff dimension of supports, existence of densities and stochastic equations.

Absolute Continuity of the Measure States in a Branching Model with Catalysts

Spatially homogeneous measure-valued branching Markov processes X on the real line ℝ with certain motion processes and branching mechanisms with finite variances have absolutely continuous states with respect to Lebesgue measure, that is, roughly speaking,

An infinite system of Brownian balls with infinite range interaction

X(t,dy) = \eta (t,y)dy

INFINITE SYSTEM OF BROWNIAN BALLS: EQUILIBRIUM MEASURES ARE CANONICAL GIBBS

for some random density function η(t)=η(t,·). Results of this type are established in Dawson and Hochberg (1979), Roelly-Coppoletta (1986), Wulfsohn (1986), Konno and Shiga (1988), and Tribe (1989).

Long time behavior of stochastic hard ball systems

The infinite particle system known as spatial branching process has been introduced (under the name Branching Markov process) by Ikeda, Nagasawa, Watanabe ([I-N-W]).

Limite ergodique de processus de diffusion infini-dimensionnels

We study an infinite system of Brownian hard balls, moving in and submitted to a smooth infinite range pair potential. It is represented by a diffusion process, which is constructed as the unique strong solution of an infinite-dimensional Skorohod equation. We also prove that canonical Gibbs states associated to the sum of the hard core potential and the pair potential are reversible measures for the dynamics.