We show that the centred occupation time process of the origin of a system of critical binary branching random walks in dimension $d\ge 3$, started off either from a Poisson field or in equilibrium, when suitably normalized, converges to a Brownian motion in $d\ge4$. In $d=3$, the limit process is a fractional Brownian motion with Hurst parameter 3/4 when starting in equilibrium, and a related Gaussian process when starting from a Poisson field. For (dependent) branching random walks with state dependent branching rate we obtain convergence in f.d.d. to the same limit process, and for $d=3$ also a functional limit theorem.
In this paper we consider two related stochastic models. The first one is a branching system consisting of particles moving according to a Markov family in R^d and undergoing subcritical branching with a constant rate of V>0. New particles immigrate to the system according to a homogeneous space time Poisson random field. The second model is the superprocess corresponding to the branching particle system. We study rescaled occupation time process and the process of its fluctuations with very mild assumptions on the Markov family. In the general setting a functional central limit theorem as wel…
In this paper we consider a branching particle system consisting of particles moving according to the Ornstein-Uhlenbeck process in $\Rd$ and undergoing a binary, supercritical branching with a constant rate $λ>0$. This system is known to fulfil a law of large numbers (under exponential scaling). In the paper we prove the corresponding central limit theorem. The limit and the CLT normalisation fall into three qualitatively different classes. In, what we call, the small branching rate case the situation resembles the classical one. The weak limit is Gaussian and normalisation is the square root…
In this paper we consider a branching particle system consisting of particles moving according to the Ornstein-Uhlenbeck process in R^d and undergoing a binary, supercritical branching with a constant rate λ>0. This system is known to fulfil a law of large numbers (under exponential scaling). In the paper we prove the corresponding central limit theorem. Moreover, in the second part of the paper we consider U-statistics of the system, for which, under mild assumptions, we prove a law of large numbers and a central limit theorem. The limits are expressed in terms of multiple stochastic integral…
Consider a branching system with particles moving according to an Ornstein-Uhlenbeck process with drift $μ>0$ and branching according to a law in the domain of attraction of the $(1+β)$-stable distribution. The mean of the branching law is strictly larger than $1$ implying that the system is supercritical and the total number of particles grows exponentially at some rate $λ>0$. It is known that the system obeys a law of large numbers. In the paper we study its rate of convergence. We discover an interesting interplay between the branching rate $λ$ and the drift parameter $μ$. There are three r…
Let $(X_1 , \ldots , X_d)$ be random variables taking nonnegative integer values and let $f(z_1, \ldots , z_d)$ be the probability generating function. Suppose that $f$ is real stable; equivalently, suppose that the polarization of this probability distribution is strong Rayleigh. In specific examples, such as occupation counts of disjoint sets by a determinantal point process, it is known~\cite{soshnikov02} that the joint distribution must approach a multivariate Gaussian distribution. We show that this conclusion follows already from stability of $f$.
Functional limit theorems are presented for the rescaled occupation time fluctuations process of a critical finite variance branching particle system in $R^d$ with symmetric a-stable motion starting off from either a standard Poisson random field or from the equilibrium distribution for intermediate dimensions a<d<2a. The limit processes are determined sub-fractional and fractional Brownian motion respectively.
The occupation time of an age-dependent branching particle system in $\Rd$ is considered, where the initial population is a Poisson random field and the particles are subject to symmetric $α$-stable migration, critical binary branching and random lifetimes. Two regimes of lifetime distributions are considered: lifetimes with finite mean and lifetimes belonging to the normal domain of attraction of a $γ$-stable law, $γ\in(0,1)$. It is shown that in dimensions $d>αγ$ for the heavy-tailed lifetimes case, and $d>α$ for finite mean lifetimes, the occupation time proccess satisfies a strong law of l…
Limit theorems are presented for the rescaled occupation time fluctuation process of a critical finite variance branching particle system in $\mathbb{R}^{d}$ with symmetric $α$-stable motion starting off from either a standard Poisson random field or the equilibrium distribution for critical $d=2α$ and large $d>2α$ dimensions. The limit processes are generalised Wiener processes. The obtained convergence is in space-time, finite-dimensional distributions sense. With the addtional assumption on the branching law we obtain functional convergence.
We study the limiting occupation density process for a large number of critical and driftless branching random walks. We show that the rescaled occupation densities of $\lfloor sN\rfloor$ branching random walks, viewed as a function-valued, increasing process $\{g_{s}^{N}\}_{s\ge 0}$, converges weakly to a pure jump process in the Skorohod space $\mathbb D([0, +\infty), \mathcal C_{0}(\mathbb R))$, as $N\to\infty$. Moreover, the jumps of the limiting process consist of i.i.d. copies of an Integrated super-Brownian Excursion (ISE) density, rescaled and weighted by the jump sizes in a real-value…
We establish limit theorems for re-scaled occupation time fluctuations of a sequence of branching particle systems in $\R^d$ 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 non-stationary 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.{\it Stochastic Pr…
We establish limit theorems involving weak convergence of multiple generations of critical and supercritical branching processes. These results arise naturally when dealing with the joint asymptotic behavior of functionals defined in terms of several generations of such processes. Applications of our main result include a functional central limit theorem (CLT), a Darling-Erdös result, and an extremal process result. The limiting process for our functional CLT is an infinite dimensional Brownian motion with sample paths in the infinite product space $(C_0[0,1])^{\infty}$, with the product topol…
In the paper the rescaled occupation time fluctuation process of a certain empirical system is investigated. The system consists of particles evolving independently according to α-stable motion in R^d, α<d<2α. The particles split according to the binary critical branching law with intensity V>0. We study how the limit behaviour of the fluctuations of the occupation time depends on the \emph{initial particle configuration}. We obtain a functional central limit theorem for a vast class of infinitely divisible distributions. Our findings extend and put in a unified setting results which previousl…
We consider a branching system consisting of particles moving according to a Markov family in $\Rd$ and undergoing subcritical branching with a constant rate $V>0$. New particles immigrate to the system according to homogeneous space-time Poisson random field. The process of the fluctuations of the rescaled occupation time is studied with very mild assumptions on the Markov family. In this general setting a functional central limit theorem is proved. The subcriticality of the branching law is crucial for the limit behaviour and in a sense overwhelms the properties of the particles' motion. It …
We establish limit theorems for the fluctuations of the rescaled occupation time of a $(d,α,β)$-branching particle system. It consists of particles moving according to a symmetric $α$-stable motion in $\mathbb{R}^d$. The branching law is in the domain of attraction of a (1+$β$)-stable law and the initial condition is an equilibrium random measure for the system (defined below). In the paper we treat separately the cases of intermediate $α/β<d<(1+β)α/β$, critical $d=(1+β)α/β$ and large $d>(1+β)α/β$ dimensions. In the most interesting case of intermediate dimensions we obtain a version of a frac…
We 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 …
We prove limit theorems for rescaled occupation time fluctuations of a (d,alpha,beta)-branching particle system (particles moving in R^d according to a spherically symmetric alpha-stable Levy process, (1+beta)-branching, 0<beta<1, uniform Poisson initial state), in the cases of critical dimension, d=alpha(1+beta)/beta, and large dimensions, d>alpha(1+beta)/beta. The fluctuation processes are continuous but their limits are stable processes with independent increments, which have jumps. The convergence is in the sense of finite-dimensional distributions, and also of space-time random fields (ti…
The $(d,α,β,γ)$-branching particle system consists of particles moving in $R^d$ according to a symmetric $α$-stable Lévy process $(0<α\leq 2)$, splitting with a critical $(1+β)$-branching law $(0<β\leq 1)$, and starting from an inhomogeneous Poisson random measure with intensity measure $μ_γ(dx)=dx/(1+|x|^γ), γ\geq 0$. By means of time rescaling $T$ and Poisson intensity measure $H_Tμ_γ$, occupation time fluctuation limits for the system as $T\to\infty$ have been obtained in two special cases: Lebesgue measure ($γ=0$, the homogeneous case), and finite measures $(γ>d)$. In some cases $H_T\equiv…
Customers arrive at rate N times alpha on a network of N single server infinite buffer queues, choose L queues uniformly, join the shortest one, and are served there in turn at rate beta. We let N go to infinity.We prove a functional central limit theorem (CLT) for the tails of the empirical measures of the queue occupations,in a Hilbert space with the weak topology, with limit given by an Ornstein-Uhlenbeck process. The a priori assumption is that the initial data converge.This completes a recent functional CLT in equilibrium result for which convergence for the initial data was not known in …
Occupation-Time Fluctuations of an Age-Dependent Branching System Driven by Poisson Immigration
Centro de Investigación en Matemáticas（数学研究中心）；Faculty of Mathematics and Informatics, Sofia University “St. Kliment Ohridski”（索非亚“圣克利门特·奥赫里德斯基”大学数学与信息学院）；Institute of Mathematics and Informatics, Bulgarian Academy of Sciences（保加利亚科学院数学与情报研究所）
We study occupancy counts for the critical nearest-neighbor branching random walk on the $d$-dimensional lattice, conditioned on non-extinction. For $d\geq 3$, Lalley and Zheng (2011) showed that the properly scaled joint distribution of the number of sites occupied by $j$ generation-$n$ particles, $j=1,2,\ldots$, converges in distribution as $n$ goes to infinity, to a deterministic multiple of a single exponential random variable. The limiting exponential variable can be understood as the classical Yaglom limit of the total population size of generation $n$. Here we study the second order flu…
Consider a critical nearest neighbor branching random walk on the $d$-dimensional integer lattice initiated by a single particle at the origin. Let $G_{n}$ be the event that the branching random walk survives to generation $n$. We obtain limit theorems conditional on the event $G_{n}$ for a variety of occupation statistics: (1) Let $V_{n}$ be the maximal number of particles at a single site at time $n$. If the offspring distribution has finite $α$th moment for some integer $α\geq 2$, then in dimensions 3 and higher, $V_n=O_p(n^{1/α})$; and if the offspring distribution has an exponentially dec…
Occupation time fluctuation limits of particle systems in R^d with independent motions (symmetric stable Levy process, with or without critical branching) have been studied assuming initial distributions given by Poisson random measures (homogeneous and some inhomogeneous cases). In this paper, with d=1 for simplicity, we extend previous results to a wide class of initial measures obeying a quasi-homogeneity property, which includes as special cases homogeneous Poisson measures and many deterministic measures (simple example: one atom at each point of Z), by means of a new unified approach. In…
We consider a branching particle system consisting of particles moving according to the Ornstein-Uhlenbeck process in $\Rd$ and undergoing a binary, supercritical branching with a constant rate $λ>0$. This system is known to fulfil a law of large numbers (under exponential scaling). Recently the question of the corresponding central limit theorem has been addressed. It turns out that the normalization and form of the limit in the CLT fall into three qualitatively different regimes, depending on the relation between the branching intensity and the parameters of the Orstein-Uhlenbeck process. In…
In this paper we analyze a branching process with immigration defined recursively by $X_t=θ_t\circ X_{t-1}+B_t$ for a sequence $(B_t)$ of i.i.d. random variables and random mappings $ θ_t\circ x:=θ_t(x)=\sum_{i=1}^xA_i^{(t)}, $ with $(A_i^{(t)})_{i\in \mathbb{N}_0}$ being a sequence of $\mathbb{N}_0$-valued i.i.d. random variables independent of $B_t$. We assume that one of generic variables $A$ and $B$ has a regularly varying tail distribution. We identify the tail behaviour of the distribution of the stationary solution $X_t$. We also prove CLT for the partial sums that could be further gene…
We study the convergence in distribution norms in the Central Limit Theorem for non identical distributed random variables that is $$ \varepsilon_{n}(f):={\mathbb{E}}\Big(f\Big(\frac 1{\sqrt n}\sum_{i=1}^{n}Z_{i}\Big)\Big)-{\mathbb{E}}\big(f(G)\big)\rightarrow 0 $$ where $Z_{i}$ are centred independent random variables and $G$ is a Gaussian random variable. We also consider local developments (Edgeworth expansion). This kind of results is well understood in the case of smooth test functions $f$. If one deals with measurable and bounded test functions (convergence in total variation distance), …
Functional limit theorems for scaled fluctuations of occupation time processes of a sequence of critical branching particle systems in $\R^d$ with anisotropic space motions and strongly degenerated splitting abilities are proved in the cases of critical and intermediate dimensions. The results show that the limit processes are constant measure-valued Wienner processes with degenerated temporal and simple spatial structures.
In branching diffusions with immigration (BDI), particles travel on independent diffusion paths in $\mathbb{R}^d$, branch at position-dependent rates and leave offspring -- randomly scattered around the parent's death position -- according to position-dependent laws. We specify a set of conditions which grants ergodicity such that the invariant occupation measure is of finite total mass and admits a continuous Lebesgue density. Under discrete-time observation, BDI configurations being recorded at discrete times $iΔ$ only, $i\in\mathbb{N}_0$, we lose information about particle identities betwee…
The myopic (or `true') self-avoiding walk model (MSAW) was introduced in the physics literature by Amit, Parisi and Peliti (1983). It is a random motion in Z^d pushed towards domains less visited in the past by a kind of negative gradient of the occupation time measure. We investigate the asymptotic behaviour of MSAW in the non-recurrent dimensions. For a wide class of self-interaction functions, we identify a natural stationary (in time) and ergodic distribution of the environment (the local time profile) as seen from the moving particle and we establish diffusive lower and upper bounds for t…
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 …
For a discrete time multitype supercritical Galton-Watson process $(Z_n)_{n\in \mathbb{N}}$ and corresponding genealogical tree $\mathbb{T}$, we associate a new discrete time process $(Z_n^Φ)_{n\in\mathbb{N}}$ such that, for each $n\in \mathbb{N}$, the contribution of each individual $u\in\mathbb{T}$ to $Z_n^Φ$ is determined by a (random) characteristic $Φ$ evaluated at the age of $u$ at time $n$. In other words, $Z_n^Φ$ is obtained by summing over all $u\in \mathbb{T}$ the corresponding contributions $Φ_u$, where $(Φ_u)_{u\in \mathbb{T}}$ are i.i.d. copies of $Φ$. Such processes are known in …
We extend the construction by Kuelske and Iacobelli of metastates in finite-state mean-field models in independent disorder to situations where the local disorder terms are are a sample of an external ergodic Markov chain in equilibrium. We show that for non-degenerate Markov chains, the structure of the theorems is analogous to the case of i.i.d. variables when the limiting weights in the metastate are expressed with the aid of a CLT for the occupation time measure of the chain. As a new phenomenon we also show in a Potts example that, for a degenerate non-reversible chain this CLT approximat…
The problems considered in the present paper have their roots in two different cultures. The 'true' (or myopic) self-avoiding walk model (TSAW) was introduced in the physics literature by Amit, Parisi and Peliti. This is a nearest neighbor non-Markovian random walk in Z^d which prefers to jump to those neighbors which were less visited in the past. The self-repelling Brownian polymer model (SRBP), initiated in the probabilistic literature by Durrett and Rogers (independently of the physics community), is the continuous space-time counterpart: a diffusion in R^d pushed by the negative gradient …
Near critical catalyst-reactant branching processes with controlled immigration are studied. The reactant population evolves according to a branching process whose branching rate is proportional to the total mass of the catalyst. The bulk catalyst evolution is that of a classical continuous time branching process; in addition there is a specific form of immigration. Immigration takes place exactly when the catalyst population falls below a certain threshold, in which case the population is instantaneously replenished to the threshold. Such models are motivated by problems in chemical kinetics …
The notion of degree and related notions concerning recurrence and transience for a class of L'evy processes on metric Abelian groups are studied. The case of random walks on a hierarchical group is examined with emphasis on the role of the ultrametric structure of the group and on analogies and differences with Euclidean random walks. Applications to separation of time scales and occupation times of multilevel branching systems are discussed.
We prove a functional limit theorem for the rescaled occupation time fluctuations of a $(d,α,β)$-branching particle system [particles moving in $\mathbb {R}^d$ according to a symmetric $α$-stable Lévy process, branching law in the domain of attraction of a $(1+β)$-stable law, $0<β<1$, uniform Poisson initial state] in the case of intermediate dimensions, $α/β<d<α(1+β)/β$. The limit is a process of the form $Kλξ$, where $K$ is a constant, $λ$ is the Lebesgue measure on $\mathbb {R}^d$, and $ξ=(ξ_t)_{t\geq0}$ is a $(1+β)$-stable process which has long range dependence. For $α<2$, there are two l…
We study solutions to the evolution equation $u_t=Δu-u +\sum_{k\geqslant 1}q_ku^k$, $t>0$, in $\mathbf{R}^d$. Here the coefficients $q_k\geqslant 0$ verify $ \sum_{k\geqslant 1}q_k=1< \sum_{k\geqslant 1}kq_k<\infty$. First, we deal with existence, uniqueness, and the asymptotic behavior of the solutions as $t\to +\infty$. We then deduce results on the long time behavior of the associated branching process, with state space the set of all finite configurations of $\mathbf{R}^d$, under the assumption that $\sum_{k\geq 1} k^2q_k<\infty$. It turns out that the distribution of the branching process…
Consider a branching random walk on $\mathbb Z$ in discrete time. Denote by $L_n(k)$ the number of particles at site $k\in\mathbb Z$ at time $n\in\mathbb N_0$. By the profile of the branching random walk (at time $n$) we mean the function $k\mapsto L_n(k)$. We establish the following asymptotic expansion of $L_n(k)$, as $n\to\infty$: $$ e^{-φ(0)n} L_n(k) = \frac{e^{-\frac 12 x_n^2(k)}}{\sqrt {2πφ''(0) n}} \sum_{j=0}^r \frac{F_j(x_n(k))}{n^{j/2}} + o\left(n^{-\frac{r+1}{2}}\right) \quad a.s., $$ where $r\in\mathbb N_0$ is arbitrary, $φ(β)=\log \sum_{k\in\mathbb Z} e^{βk} \mathbb E L_1(k)$ is th…
The self-repelling Brownian polymer model (SRBP) initiated by Durrett and Rogers in [Durrett-Rogers (1992)] is the continuous space-time counterpart of the myopic (or 'true') self-avoiding walk model (MSAW) introduced in the physics literature by Amit, Parisi and Peliti in [Amit-Parisi-Peliti (1983)]. In both cases, a random motion in space is pushed towards domains less visited in the past by a kind of negative gradient of the occupation time measure. We investigate the asymptotic behaviour of SRBP in the non-recurrent dimensions. First, extending 1-dimensional results from [Tarres-Toth-Valko…
