Self-similar stable processes arising from high-density limits of occupation times of particle systems

Abstract

We extend results on time-rescaled occupation time fluctuation limits of the $(d,α, β)$-branching particle system $(0<α\leq 2, 0<β\leq 1)$ with Poisson initial condition. The earlier results in the homogeneous case (i.e., with Lebesgue initial intensity measure) were obtained for dimensions $d>α/ β$ only, since the particle system becomes locally extinct if $d\le α/ β$. In this paper we show that by introducing high density of the initial Poisson configuration, limits are obtained for all dimensions, and they coincide with the previous ones if $d>α/β$. We also give high-density limits for the systems with finite intensity measures (without high density no limits exist in this case due to extinction); the results are different and harder to obtain due to the non-invariance of the measure for the particle motion. In both cases, i.e., Lebesgue and finite intensity measures, for low dimensions ($d<α(1+β)/β$ and $d<α(2+β)/(1+β)$, respectively) the limits are determined by non-Lévy self-similar stable processes. For the corresponding high dimensions the limits are qualitatively different: ${\cal S}'(R^d)$-valued Lévy processes in the Lebesgue case, stable processes constant in time on $(0,\infty)$ in the finite measure case. For high dimensions, the laws of all limit processes are expressed in terms of Riesz potentials. If $β=1$, the limits are Gaussian. Limits are also given for particle systems without branching, which yields in particular weighted fractional Brownian motions in low dimensions. The results are obtained in the setup of weak convergence of S'(R^d)$-valued processes.