Limit theorems for occupation time fluctuations of branching systems II: Critical and large dimensions

We give functional limit theorems for the fluctuations of the rescaled occupation time process of a critical branching particle system in R^d${\mathbb{R}}^d$ with symmetric α$\alpha$-stable motion in the cases of critical and large dimensions, d=2α$d=2\alpha$ and d>2α$d>2\alpha$. In a previous paper [T. Bojdecki, L.G. Gorostiza, A. Talarczyk, Limit theorems for occupation time fluctuations of branching systems I: long-range dependence, Stochastic Process. Appl., this issue.] we treated the case of intermediate dimensions, α<d<2α$\alpha <d<2\alpha$, which leads to a long-range dependence limit process. In contrast, in the present cases the limits are generalized Wiener processes. We use the same space–time random field method of the previous paper, the main difference being that now the tightness requires a new approach and the proofs are more difficult. We also give analogous results for the system without branching in the cases d=α$d=\alpha$ and d>α$d>\alpha$.     

Particle picture interpretation of some Gaussian processes related to fractional Brownian motion

We construct fractional Brownian motion, sub-fractional Brownian motion and negative sub-fractional Brownian motion by means of limiting procedures applied to some particle systems. These processes are obtained for full ranges of Hurst parameter.     

Limit theorems for occupation time fluctuations of branching systems I: Long-range dependence

We give a functional limit theorem for the fluctuations of the rescaled occupation time process of a critical branching particle system in R^d${\mathbb{R}}^d$ with symmetric α$\alpha$-stable motion and α<d<2α$\alpha <d<2\alpha$, which leads to a long-range dependence process involving sub-fractional Brownian motion. We also give an analogous result for the system without branching and d<α$d<\alpha$, which involves fractional Brownian motion. We use a space–time random field approach.     

Self-Similar Stable Processes Arising from High-Density Limits of Occupation Times of Particle Systems

We extend results on time-rescaled occupation time fluctuation limits of the (d, α, β)-branching particle system (0 < α ≤ 2, 0 < β ≤ 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 ≤ 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: -valued Lévy processes in the Lebesgue case, stable processes constant in time on (0,∞) 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 -valued processes. Research supported by MNiSW grant 1P03A1129 (Poland; T. Bojdecki and A. Talarczyk) and by CONACyT grant 45684-F (Mexico; L.G. Gorostiza).     

Self-Intersection Local Time for Some S′(ℝd)-Ornstein-Uhlenbeck Processes Related to Inhomogeneous Fields

We study existence and path continuity of the self-intersection local time (SILT) for some S′(ℝ^d)-valued Ornstein-Uhlenbeck processes. Examples of such processes arise in particular as fluctuation limits ofparticle systems. We analyze the effect that different types ofmeasures involved in the covariances of the processes have on existence and continuity of SILT. These measures do not necessarily have the regularity and homogeneity properties of those that have been considered before; this precludes some key ingredients of the previous techniques. We develop new technical tools and prove sufficient conditions and some necessary conditions for existence of SILT, and sufficient conditions for path continuity of SILT. The questions of existence and continuity involve problems of existence of integrals and singular integrals. We give examples which illustrate how different types of measures e.g., atomic or with L²-density) may produce different critical dimensions for existence of SILT. One of our motivations is the desire for a better understanding ofwhat SILT represents for S′(ℝ^d)-valued processes.     

Self-intersection local time for Gaussian ′( )-processes: Existence, path continuity and examples

In this paper we develop a criterion for existence or non-existence of self-intersection local time (SILT) for a wide class of Gaussian ′( ^d)-valued processes, we show that quite generally the SILT process has continuous paths, and we give several examples which illustrate existence of SILT for different ranges of dimensions (e.g., d ≤ 3, d ≤ 7 and 5 ≤ d ≤ 11 in the Brownian case). Some of the examples involve branching and exhibit “dimension gaps”. Our results generalize the work of Adler and coauthors, who studied the special case of “density processes” and proved that SILT paths are cadlag in the Brownian case making use of a “particle picture” approximation (this technique is not available for our general formulation).     

Time-Localization of Random Distributions on Wiener Space

A nuclear space of distributions on Wiener space was constructed by Gorostiza and Nualart [10] as a framework for studying weak convergence of trajectorial fluctuations of particle systems. A basic problem in recovering the usual time-evolution results from the trajectorial ones consists in associating in a unique way an -valued process to a random distribution on by localizing it at each time t [0,1]. In this paper we solve this problem for a large class of random distributions which includes trajectorial fluctuation limits of some systems of diffusions.     

Self‐Intersection Local Time for 𝒮′(ℝd)‐Ornstein‐Uhlenbeck Processes Arising from Immigration Systems

Fluctuation limits of an immigration branching particle system and an immigration branching measure‐valued process yield different types of 𝒮′(ℝ^d)‐valued Ornstein‐Uhlenbeck processes whose covariances are given in terms of an excessive measure for the underlying motion in Rd, which is taken to be a symmetric α‐stable process. In this paper we prove existence and path continuity results for the self‐intersection local time of these Ornstein‐Uhlenbeck processes. The results depend on relationships between the dimension d and the parameter α.     

Oscillatory Fractional Brownian Motion

We introduce oscillatory analogues of fractional Brownian motion, sub-fractional Brownian motion and other related long range dependent Gaussian processes, we discuss their properties, and we show how they arise from particle systems with or without branching and with different types of initial conditions, where the individual particle motion is the so-called c-random walk on a hierarchical group. The oscillations are caused by the ultrametric structure of the hierarchical group, and they become slower as time tends to infinity and faster as time approaches zero. A randomness property of the initial condition increases the long range dependence. We emphasize the new phenomena that are caused by the ultrametric structure as compared with results for analogous models on Euclidean space.