Limit Theorems for Supercritical Branching Random Fields

An infinite particle system in R^d is considered where the initial distribution is POISSON ian and each initial particle gives rise to a supercritical age-dependent branching process with the particles moving randomly in space. Our approach differs from the usual: instead of the point measures determined by the locations of the particles at each time, we take the particles at a “final time” and observe the past histories of their ancestry lines. A law of large numbers and a central limit theorem are proved under a space-time scaling representing high density of particles and small mean particle lifetime. The fluctuation limit is a generalized GAUSS -MARKOV process with continuous trajectories and satisfies a deterministic evolution equation with generalized random initial condition. A more precise form of the central limit theorem is obtained in the case of particles performing BROWN ian motion and having exponentially distributed lifetime.     

Multi-scale Clustering for a Non-Markovian Spatial Branching Process

Consider a system of particles which move in R^d according to a symmetric α-stable motion, have a lifetime distribution of finite mean, and branch with an offspring law of index 1+β. In case of the critical dimension d=α/β the phenomenon of multi-scale clustering occurs. This is expressed in an fdd scaling limit theorem, where initially we start with an increasing localized population or with an increasing homogeneous Poissonian population. The limit state is uniform, but its intensity varies in line with the scaling index according to a continuous-state branching process of index 1+β. Our result generalizes the case α=2 of Brownian particles of Klenke (1998), where p.d.e. methods had been used which are not available in the present setting. Supported in part by the DFG. Supported in part by the grants RFBR 02-01-00266 and Russian Scientific School 1758.2003.1.     

Convergence to equilibrium of critical branching particle systems and superprocesses,and related nonlinear partial differential equations

We consider a class of multitype particle systems in ^d undergoing spatial diffusion and critical stable multitype branching, and their limits known as critical stable multitype Dawson-Watanabe processes, or superprocesses. We show that for large classes of initial states, the particle process and the superprocess converge in distribution towards known equilibrium states as time tends to infinity. As an application we obtain the asymptotic behavior of a system of nonlinear partial differential equations whose solution is related to the distribution of both the particle process and the superprocess.Research partially supported by CONACyT (Mexico), CNRS (France) and BMfWuF (Austria).     

Limit Theorems for Branching Diffusions in Hydrodynamical Rescaling

The hydrodynamical limit is studied for infinite systems of BROWN ian particles in R^d, d = 3, which branch out at exponentially distributed times according to a critical offspring distribution with finite second moment. In the second part a central limit theorem for the hydrodynamical fluctuations is derived in the case, when the branching mechanism has a finite third moment.     

Persistence of critical multitype particle and measure branching processes

Summary We consider a class of systems of particles ofk types inR ^d 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.Research partially supported by CONACyT (Mexico), CNRS (France) and BMfWuF (Austria).     

Superprocesses with Coalescing Brownian Spatial Motion as Large-Scale Limits

A Superprocess with coalescing spatial motion is constructed in terms of one-dimensional excursions. Based on this construction, it is proved that the superprocess is purely atomic and arises as scaling limit of a special form of the superprocess with dependent spatial motion studied in Dawson et al. (Refs. 5, 19–20).     

Occupation Time Fluctuations of Weakly Degenerate Branching Systems

We establish limit theorems for rescaled occupation time fluctuations of a sequence of branching particle systems in ? ^d with anisotropic space motion and weakly degenerate splitting ability. In the case of large dimensions, our limit processes lead to a new class of operator-scaling Gaussian random fields with nonstationary increments. In the intermediate and critical dimensions, the limit processes have spatial structures analogous to (but more complicated than) those arising from the critical branching particle system without degeneration considered by Bojdecki et?al. (Stoch. Process. Appl. 116:1?C18 and 19?C35, 2006). Due to the weakly degenerate branching ability, temporal structures of the limit processes in all three cases are different from those obtained by Bojdecki et?al. (Stoch. Process. Appl. 116:1?C18 and 19?C35, 2006).     

Large deviations for the symmetric simple exclusion process in dimensions d≥ 3

We consider symmetric simple exclusion processes with L=&ρmacr;N ^d particles in a periodic d-dimensional lattice of width N. We perform the diffusive hydrodynamic scaling of space and time. The initial condition is arbitrary and is typically far away form equilibrium. It specifies in the scaling limit a density profile on the d-dimensional torus. We are interested in the large deviations of the empirical process, N ^{− d} [∑ ^L ₁δ _{xi (·)}] as random variables taking values in the space of measures on D[0.1]. We prove a large deviation principle, with a rate function that is more or less universal, involving explicity besides the initial profile, only such canonical objects as bulk and self diffusion coefficients. Received: 7 September 1997 / Revised version: 15 May 1998     

Rate of growth of the coalescent set in a coalescing stochastic flow

Consider an isotropic stochastic flow in R_d (i.e. a simultaneous random, correlated motion of all points in space), where d=l,2 or 3, such that the joint law of the motion of two particles allows the particles to meet and coalesce in finite time. The coalescent set J _t is a random subset of R_d consisting of the initial positions of particles which have coalesced by time t with the particle which started at 0. We show that the expected volume of J t grows at a rate proportional to when d=1, and at rates close to proportional to t/log t (resp. t) when d = 2 (resp. d=3). We give an example of a coalescing stochastic flow when d = 3. These results are analogous to growth rates of expected population size of a surviving type in the "invasion process" described by Clifford and Sudbury     

Semi-Selfsimilar Processes

A notion of semi-selfsimilarity of R ^d -valued stochastic processes is introduced as a natural extension of the selfsimilarity. Several topics on semi-selfsimilar processes are studied: the existence of the exponent for semi-selfsimilar processes; characterization of semi-selfsimilar processes as scaling limits; relationship between semi-selfsimilar processes with independent increments and semi-selfdecomposable distributions, and examples; construction of semi-selfsimilar processes with stationary increments; and extension of the Lamperti transformation. Semi-stable processes where all joint distributions are multivariate semi-stable are also discussed in connection with semi-selfsimilar processes. A wide-sense semi-selfsimilarity is defined and shown to be reducible to semi-selfsimilarity.     

A weighted occupation time for a class of measured-valued branching processes

Summary A weighted occupation time is defined for measure-valued processes and a representation for it is obtained for a class of measure-valued branching random motions on R ^d. Considered as a process in its own right, the first and second order asymptotics are found as time t. Specifically the finiteness of the total weighted occupation time is determined as a function of the dimension d, and when infinite, a central limit type renormalization is considered, yielding Gaussian or asymmetric stable generalized random fields in the limit. In one Gaussian case the results are contrasted in high versus low dimensions.Research supported in part by Natural Sciences and Engineering Research Council of Canada     

Spread of branching diffusion: Sharp asymptotics

A one-dimensional branching diffusion with a stable drift is considered. Let R_t denote the position of the rightmost particle at time t. It is shown that R_t , properly normalized, has a limiting distribution.     

Scaling limits for internal aggregation models with multiple sources

We study the scaling limits of three different aggregation models on ℤ ^d : internal DLA, in which particles perform random walks until reaching an unoccupied site; the rotor-router model, in which particles perform deterministic analogues of random walks; and the divisible sandpile, in which each site distributes its excess mass equally among its neighbors. As the lattice spacing tends to zero, all three models are found to have the same scaling limit, which we describe as the solution to a certain PDE free boundary problem in ℝ ^d . In particular, internal DLA has a deterministic scaling limit. We find that the scaling limits are quadrature domains, which have arisen independently in many fields such as potential theory and fluid dynamics. Our results apply both to the case of multiple point sources and to the Diaconis-Fulton smash sum of domains.     

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.     

Limit distributions arising in branching random walks on integer lattices

The paper completes the investigation of limit distribution of the number of particles at the source of branching in the model of critical catalytic branching random walk on ^dd N {{\mathbb Z}^d}\;d \in {\mathbb N} . Limit theorems of such kind were established only for d = 1, 2, 3, 4 under the assumption that, at the initial moment, there is a single particle at the source of branching. We prove their analog for d \geqslant 5 d \geqslant 5 . Moreover, in any dimension, we generalize the previous results by permitting the initial particle to start at an arbitrary point of the lattice.     

The Exact Hausdorff Measure Function of the Level Sets of Multi-parameter Symmetric Stable Process

In this paper, we investigate the Hausdorff measure for level sets of N-parameter Rd-valued stable processes, and develop a means of seeking the exact Hausdorff measure function for level sets of N-parameter Rd-valued stable processes. We show that the exact Hausdorff measure function of level sets of N-parameter Rd-valued symmetric stable processes of index α is Ф（r） = r^N-d/α （log log l/r）d/α when Nα 〉 d. In addition, we obtain a sharp lower bound for the Hausdorff measure of level sets of general （N, d, α） strictly stable processes.     

An almost sure scaling limit theorem for Dawson-Watanabe superprocesses

We establish a scaling limit theorem for a large class of Dawson-Watanabe superprocesses whose underlying spatial motions are symmetric Hunt processes, where the convergence is in the sense of convergence in probability. When the underling process is a symmetric diffusion with -coefficients or a symmetric Lévy process on Rd whose Lévy exponent Ψ(η) is bounded from below by cα|η| for some c>0 and α∈(0,2) when |η| is large, a stronger almost sure limit theorem is established for the superprocess. Our approach uses the principal eigenvalue and the ground state for some associated Schrödinger operator. The limit theorems are established under the assumption that an associated Schrödinger operator has a spectral gap.     

Measure diffusions and related explosion problems

For random measure-valued stochastic partial differential equations for biological processes, growth represented by scalar partial differential equations at each point of the support and spread being a diffusion on R ^d, solutions are constructed by smearing the growth processes at each spatial point and composing the resulting generator with the generator for the spread. If these solutions are unique the equation is called solvable. We find conditions for the noise term of a solvable equations to have trivial effect and we identify some non-solvable equations, for example the diffusion-free bilinear equation. The search led to an investigation of explosion and the effect of point barriers for scalar stochastic differential equations with linear drift; this is used to explain the clustering effect in the usual superprocess.