super-Brownian Motion Research Articles

New large deviation results for some super-Brownian processes

We give large deviation results for the super-Brownian excursion conditioned to have unit mass or unit extinction time and for super-Brownian motion with constant non-positive drift. We use a representation of these processes by a path-valued process, the so-called Brownian snake for which we state large deviation principles.

Hitting probability of a distant point for the voter model started with a single 1

The goal of this work is to find the asymptotics of the hitting probability of a distant point for the voter model on the integer lattice started from a single 1 at the origin. In dimensions d = 2 or 3, we obtain the precise asymptotic behavior of this probability. We use the scaling limit of the voter model started from a single 1 at the origin in terms of super-Brownian motion under its excursion measure. This invariance principle was stated by Bramson, Cox and Le Gall, as a consequence of a theorem of Cox, Durrett and Perkins. Less precise estimates are derived in dimension d ≥ 4.

Spatial epidemics: critical behavior in one dimension

In the simple mean-field SIS and SIR epidemic models, infection is transmitted from infectious to susceptible members of a finite population by independent p-coin tosses. Spatial variants of these models are considered, in which finite populations of size N are situated at the sites of a lattice and infectious contacts are limited to individuals at neighboring sites. Scaling laws for these models are given when the infection parameter p is such that the epidemics are critical. It is shown that in all cases there is a critical threshold for the numbers initially infected: below the threshold, the epidemic evolves in essentially the same manner as its branching envelope, but at the threshold evolves like a branching process with a size-dependent drift. The corresponding scaling limits are super-Brownian motions and Dawson–Watanabe processes with killing, respectively.

Renormalization of the two-dimensional Lotka–Volterra model

We show that renormalized two-dimensional Lotka–Volterra models near criticality converge to a super-Brownian motion. This is used to establish long-term survival of a rare type for a range of parameter values near the voter model.

Moderate deviations for the quenched mean of the super-Brownian motion with random immigration

Moderate deviations for the quenched mean of the super-Brownian motion with random immigration are proved for 3 ⩽ d ⩽ 6, which fills in the gap between central limit theorem (CLT) and large deviation principle (LDP).

A LARGE DEVIATION FOR OCCUPATION TIME OF SUPER α-STABLE PROCESS

We derive a large deviation principle for occupation time of super α-stable process in ℝd with d > 2α. The decay of tail probabilities is shown to be exponential and the rate function is characterized. Our result can be considered as a counterpart of Lee's work on large deviations for occupation times of super-Brownian motion in ℝd for dimension d > 4 (see Ref. 10).

Some scaled limit theorems for an immigration super-Brownian motion

In this paper, the small time limit behaviors for an immigration super-Brownian motion are studied, where the immigration is determined by Lebesgue measure. We first prove a functional central limit theorem, and then study the large and moderate deviations associated with this central tendency.

Understanding Clustering in Type Space Using Field Theoretic Techniques

The birth/death process with mutation describes the evolution of a population, and displays rich dynamics including clustering and fluctuations. We discuss an analytical 'field-theoretical' approach to the birth/death process, using a simple dimensional analysis argument to describe evolution as a 'super-Brownian motion' in the infinite population limit. The field theory technique provides corrections to this for large but finite population, and an exact description at arbitrary population size. This allows a characterisation of the difference between the evolution of a phenotype, for which strong local clustering is observed, and a genotype for which distributions are more dispersed. We describe the approach with sufficient detail for non-specialists.

A zero–one law of almost sure local extinction for [formula omitted]-super-Brownian motion

This paper considers the following generalized almost sure local extinction for the d -dimensional ( 1 + β ) -super-Brownian motion X starting from Lebesgue measure on R d . For any t ≥ 0 write B g ( t ) for a closed ball in R d with center at 0 and radius g ( t ) , where g is a nonnegative, nondecreasing and right continuous function on [ 0 , ∞ ) . Let τ ≔ sup { t ≥ 0 : X t ( B g ( t ) ) > 0 } . For d β < 2 , it is shown that P { τ = ∞ } is equal to either 0 or 1 depending on whether the value of the integral ∫ 1 ∞ g ( y ) d y − 1 − 1 / β d y is finite or infinite, respectively. An asymptotic upper bound for P { τ > t } is found when P { τ < ∞ } = 1 .

Local theorems related to Lévy-type branching mechanism

We prove local limit theorems for the total mass processes of two branching- uctuating particle systems which converge to discontinuous (2; d; fl)-superprocess. To this end, we establish new subtle properties of the total mass for this class of superpro- cesses. Thus, the density of its absolutely continuous component exhibits a polynomial blow-up at the origin and has a regularly varying upper tail. Both particle systems consid- ered are characterized by the same heavy-tailed branching mechanism that belongs to the domain of normal attraction of an extreme stable law with index 1 + fl 2 (1; 2). One of them starts from a Poisson eld, whereas the initial number of particles for the other sys- tem is non-random. We demonstrate that the poissonization of the initial eld of particles is related to Gnedenko's method of accompanying innitely divisible laws. The compar- ison of our results with their 'continuous' counterparts (which pertain to convergence to the super-Brownian motion) reveals a worse discrepancy between the extinction proba- bilities. This is explained through the intrinsic difference between structures of individual surviving clusters.

Weak convergence of measure-valued processes and r-point functions

We prove a sufficient set of conditions for a sequence of finite measures on the space of cadlag measure-valued paths to converge to the canonical measure of super-Brownian motion in the sense of convergence of finite-dimensional distributions. The conditions are convergence of the Fourier transform of the $r$-point functions and perhaps convergence of the ``survival probabilities.'' These conditions have recently been shown to hold for a variety of statistical mechanical models, including critical oriented percolation, the critical contact process and lattice trees at criticality, all above their respective critical dimensions.

A Quenched CLT for Super-Brownian Motion with Random Immigration

A quenched central limit theorem is derived for the super-Brownian motion with super-Brownian immigration, in dimension d≥4. At the critical dimension d=4, the quenched and annealed fluctuations are of the same order but are not equal.

Feller Property and Infinitesimal Generator of the Exploration Process

We consider the exploration process associated to the continuous random tree (CRT) built using a Levy process with no negative jumps. This process has been studied by Duquesne, Le Gall and Le Jan. This measure-valued Markov process is a useful tool to study CRT as well as super-Brownian motion with general branching mechanism. In this paper we prove this process is Feller, and we compute its infinitesimal generator on exponential functionals and give the corresponding martingale.

The large scale behavior of super-Brownian motion in three dimensions with a single point source

The large scale behavior of super-Brownian motion in three dimensions with a single point source

Boundary Preserving Semianalytic Numerical Algorithms for Stochastic Differential Equations

Construction of splitting-step methods and properties of related nonnegativity and boundary preserving semianalytic numerical algorithms for solving stochastic differential equations (SDEs) of Ito type are discussed. As the crucial assumption, we oppose conditions such that one can decompose the original system of SDEs into subsystems for which one knows either the exact solution or its conditional transition probability. We present convergence proofs for a newly designed splitting-step algorithm and simulation studies for numerous well-known numerical examples ranging from stochastic dynamics occurring in asset pricing in mathematical finance (Cox-Ingersoll-Ross (CIR) and constant elasticity of variance (CEV) models) to measure-valued diffusion and super-Brownian motion (stochastic PDEs (SPDEs)) as met in biology and physics.

Local extinction for superprocesses in random environments

We consider a superprocess in a random environment represented by a random measure which is white in time and colored in space with correlation kernel $g(x,y)$. Suppose that $g(x,y)$ decays at a rate of $|x-y|^{-\alpha}$, $0\leq \alpha\leq 2$, as $|x-y|\to\infty$. We show that the process, starting from Lebesgue measure, suffers longterm local extinction. If $\alpha < 2$, then it even suffers finite time local extinction. This property is in contrast with the classical super-Brownian motion which has a non-trivial limit when the spatial dimension is higher than 2. We also show in this paper that in dimensions $d=1,2$ superprocess in random environment suffers local extinction for any bounded function $g$.

Survival and coexistence in stochastic spatial Lotka–Volterra models

A spatially explicit, stochastic Lotka–Volterra model was introduced by Neuhauser and Pacala in Neuhauser and Pacala (Ann. Appl. Probab. 9, 1226–1259, 1999). A low density limit theorem for this process was proved by the authors in Cox and Perkins (Ann. Probab. 33, 904–947, 2005), showing that certain generalized rescaled Lotka–Volterra models converge to super-Brownian motion with drift. Here we use this convergence result to extend what is known about the parameter regions for the Lotka–Volterra process where (i) survival of one type holds, and (ii) coexistence holds.

Pathwise Convergence of a Rescaled Super-Brownian Catalyst Reactant Process

Consider the one-dimensional catalytic super-Brownian motion X (called the reactant) in the catalytic medium $$\varrho$$ which is an autonomous classical super-Brownian motion. We characterize $$(\varrho ,X)$$ both in terms of a martingale problem and (in dimension one) as solution of a certain stochastic partial differential equation. The focus of this paper is for dimension one the analysis of the longtime behavior via a mass-time-space rescaling. When scaling time by a factor of K, space is scaled by K η and mass by K −η. We show that for every parameter value η ≥ 0 the rescaled processes converge as K→ ∞ in path space. While the catalyst’s limiting process exhibits a phase transition at η = 1, the reactant’s limit is always the same degenerate process.

Limit theorems for the weighted occupation time for super-Brownian motions on Hd

Abstract Weighted occupation time processes associated with the super-Brownian motions on hyperbolic spaces are investigated. Longtime behavior and a central limit type theorem for the processes are obtained. For all dimension d ≥ 2, the right normings for two fluctuations are t and t1/2, and the limit results are the Riemannian measure and some centered Gaussian random measure, respectively. *Supported by the Natural Science Research Program of Jiangsu Provincial Universities (Grant No. 05KJD110145)

The Burgers superprocess

We define the Burgers superprocess to be the solution of the stochastic partial differential equation ∂ ∂ t u ( t , x ) = Δ u ( t , x ) − λ u ( t , x ) ∇ u ( t , x ) + γ u ( t , x ) W ( d t , d x ) , where t ≥ 0 , x ∈ R , and W is space-time white noise. Taking γ = 0 gives the classic Burgers equation, an important, non-linear, partial differential equation. Taking λ = 0 gives the super-Brownian motion, an important, measure valued, stochastic process. The combination gives a new process which can be viewed as a superprocess with singular interactions. We prove the existence of a solution to this equation and its Hölder continuity, and discuss (but cannot prove) uniqueness of the solution.