Measure-valued Branching Processes Research Articles

Long-Time Behavior for Subcritical Measure-Valued Branching Processes with Immigration

AbstractIn this work we study the long-time behavior for subcritical measure-valued branching processes with immigration on the space of tempered measures. Under some reasonable assumptions on the spatial motion, the branching and immigration mechanisms, we prove the existence and uniqueness of an invariant probability measure for the corresponding Markov transition semigroup. Moreover, we show that it converges with exponential rate to the unique invariant measure in the Wasserstein distance as well as in a distance defined in terms of Laplace transforms. Finally, we consider an application of our results to super-Lévy processes as well as branching particle systems on the lattice with noncompact spins.

Stochastic solutions to evolution equations of non-local branching processes

We give a probabilistic representation for the solution to a nonlinear evolution equation induced by a measure-valued branching process. We first construct the involved branching processes on the set of all finite configurations of a given set, with a killing rate induced by a continuous additive functional, and with a non-local branching procedure given by a sequence of Markovian kernels. The main application is to prove stochastic aspects for a nonlinear evolution equation related to the Neumann problem and the surface measure on the boundary, which corresponds to the reflecting Brownian motion as base movement, taking the killing rate given by the local time on the boundary. We use specific potential theoretical tools.

Survival of some measure-valued Markov branching processes

We investigate conditions for survival in the L1-norm sense of the Log-Laplace equations of a class of Markov branching processes with values in the space of Radon measures on a locally compact space D. We apply our results to certain -valued superprocesses with -branching.

Numerical Solution for the Non-linear Dirichlet Problem of a Branching Process

We give a probabilistic numerical approach for the nonlinear Dirichlet problem associated with a branching process. Main tools are the probabilistic representation of the solution with the measure-valued branching process, as well as specific techniques for the numerical solution of linear partial differential equations, introduced and developed by Milstein and Tretyakov, and Monte Carlo methods.

Measure-valued branching processes associated with Neumann nonlinear semiflows

We construct a measure-valued branching Markov process associated with a nonlinear boundary value problem, where the boundary condition has a nonlinear pseudo monotone branching mechanism term −β, which includes the case β(u)=−um, with 0<m<1. The process is then used in the probabilistic representation of the solution of the parabolic problem associated with a nonlinear Neumann boundary value problem. In this way the classical association of the superprocesses to the Dirichlet boundary value problems also holds for the nonlinear Neumann boundary value problems. It turns out that the obtained branching process behaves on the measures carried by the given open set like the linear continuous semiflow, induced by the reflected Brownian motion, while the branching occurs on the measures having non-zero traces on the boundary of the open set, with the behavior of the (−β)-superprocess, having as spatial motion the process on the boundary associated to the reflected Brownian motion.

Bounded and [formula omitted]-weak solutions for nonlinear equations of measure-valued branching processes

We study the nonlinear Schrödinger type equation (Δ−c)u+c∑k=1∞qkuk=f, where c is a bounded positive function. We may replace the Laplace operator with the generator of a Borel right Markov process with state space E and consider a nonlinear term induced by a perturbing kernel. We solve this type of equation in an Lp-weak sense, involving positive definite extensions of the solution from E to the set of all finite positive measures on E. The extensions to the set of measures are related to the classical connection (from the works of M. Nagasawa, N. Ikeda, S. Watanabe, and M.L. Silverstein) between this nonlinear equation and the discrete branching Markov processes. We also investigate the existence of the bounded solutions for the Dirichlet problem associated to the above equation, with bounded boundary data. Our approach is based on probabilistic and analytic potential theoretical methods, used on both spaces E and the set of all finite configurations of E, like the Revuz formula for the continuous additive functionals and the perturbation with kernels of the transition functions and resolvents.

Stationary distributions for a class of generalized Fleming–Viot processes

We identify stationary distributions of generalized Fleming-Viot processes with jump mechanisms specified by certain beta laws together with a parameter measure. Each of these distributions is obtained from normalized stable random measures after a suitable biased transformation followed by mixing by the law of a Dirichlet random measure with the same parameter measure. The calculations are based primarily on the well-known relationship to measure-valued branching processes with immigration.

Ergodic properties for $\alpha$-CIR models and a class of generalized Fleming-Viot processes

We discuss a Markov jump process regarded as a variant of the CIR (Cox-Ingersoll-Ross) model and its infinite-dimensional extension. These models belong to a class of measure-valued branching processes with immigration, whose jump mechanisms are governed by certain stable laws. The main result gives a lower spectral gap estimate for the generator. As an application, a certain ergodic property is shown for the generalized Fleming-Viot process obtained as the time-changed ratio process.

Poisson representations of branching Markov and measure-valued branching processes

Representations of branching Markov processes and their measure-valued limits in terms of countable systems of particles are constructed for models with spatially varying birth and death rates. Each particle has a location and a "level," but unlike earlier constructions, the levels change with time. In fact, death of a particle occurs only when the level of the particle crosses a specified level $r$, or for the limiting models, hits infinity. For branching Markov processes, at each time $t$, conditioned on the state of the process, the levels are independent and uniformly distributed on $[0,r]$. For the limiting measure-valued process, at each time $t$, the joint distribution of locations and levels is conditionally Poisson distributed with mean measure $K(t)\times\varLambda$, where $\varLambda$ denotes Lebesgue measure, and $K$ is the desired measure-valued process. The representation simplifies or gives alternative proofs for a variety of calculations and results including conditioning on extinction or nonextinction, Harris's convergence theorem for supercritical branching processes, and diffusion approximations for processes in random environments.

Applications of Compact Superharmonic Functions: Path Regularity and Tightness of Capacities

We establish relations between the existence of the \({\mathcal{L}}\)-superharmonic functions that have compact level sets (\({\mathcal{L}}\) being the generator of a right Markov process), the path regularity of the process, and the tightness of the induced capacities. We present several examples in infinite dimensional situations, like the case when \({\mathcal{L}}\) is the Gross–Laplace operator on an abstract Wiener space and a class of measure-valued branching process associated with a nonlinear perturbation of \({\mathcal{L}}\).

Local Time and a Tanaka Formula for a Class of Discontinuous Measure-Valued Branching Processes

We define the local time of a discontinuous superprocess X in ℝ d with symmetric α-stable motions and (1 + β)-branching as an L 1-limit of approximating local times ∫0 t Xs (ϕ ϵ)ds, where {ϕϵ} is a sequence of smooth functions converging to δ0 in distributional sense as ϵ → 0. We prove that the limit Lt 0 exists if d < (1 + 1/β)α and does not depend of the particular choice of sequence {ϕϵ}. We show that Lt 0 admits a Tanaka formula-like representation which includes a term that incorporates the discontinuities of X. Fleischmann [10] proved that the occupation measure induced by X is absolutely continuous with respect to Lebesgue measure; using the Tanaka formula we give an easy proof of Fleischmann's result.

Competing Species Superprocesses with Infinite Variance

We study pairs of interacting measure-valued branching processes (superprocesses) with alpha-stable migration and $(1+\beta)$-branching mechanism. The interaction is realized via some killing procedure. The collision local time for such processes is constructed as a limit of approximating collision local times. For certain dimensions this convergence holds uniformly over all pairs of such interacting superprocesses. We use this uniformity to prove existence of a solution to a competing species martingale problem under a natural dimension restriction. The competing species model describes the evolution of two populations where individuals of different types may kill each other if they collide. In the case of Brownian migration and finite variance branching, the model was introduced by Evans and Perkins (1994). The fact that now the branching mechanism does not have finite variance requires the development of new methods for handling the collision local time which we believe are of some independent interest.

Skew Convolution Semigroups and Related Immigration Processes

A special type of immigration associated with measure-valued branching processes is formulated by using skew convolution semigroups. We give a characterization for a general inhomogeneous skew convolution semigroup in terms of probability entrance laws. The related immigration process is constructed by summing up measure-valued paths in the Kuznetsov process determined by an entrance rule. The behavior of the Kuznetsov process is then studied, which provides insight into trajectory structures of the immigration process. Some well-known results on excessive measures are formulated in terms of stationary immigration processes.

Multiple Scale Analysis of Spatial Branching Processes under the Palm Distribution

We consider two types of measure-valued branching processes on the lattice $Z^d$. These are on the one hand side a particle system, called branching random walk, and on the other hand its continuous mass analogue, a system of interacting diffusions also called super random walk. It is known that the long-term behavior differs sharply in low and high dimensions: if $d\leq 2$ one gets local extinction, while, for $d\geq 3$, the systems tend to a non-trivial equilibrium. Due to Kallenberg's criterion, local extinction goes along with clumping around a 'typical surviving particle.' This phenomenon is called clustering. A detailed description of the clusters has been given for the corresponding processes on $R^2$ in Klenke (1997). Klenke proved that with the right scaling the mean number of particles over certain blocks are asymptotically jointly distributed like marginals of a system of coupled Feller diffusions, called system of tree indexed Feller diffusions, provided that the initial intensity is appropriately increased to counteract the local extinction. The present paper takes different remedy against the local extinction allowing also for state-dependent branching mechanisms. Instead of increasing the initial intensity, the systems are described under the Palm distribution. It will turn out together with the results in Klenke (1997) that the change to the Palm measure and the multiple scale analysis commute, as $t\to\infty$. The method of proof is based on the fact that the tree indexed systems of the branching processes and of the diffusions in the limit are completely characterized by all their moments. We develop a machinery to describe the space-time moments of the superprocess effectively and explicitly.

Measure-valued branching processes associated with random walks on $p$-adics

Measure-valued branching random walks (superprocesses) on $p$-adics are introduced and investigated. The uniqueness and existence of solutions to associated linear and nonlinear heat-type (parabolic) equations are proved, provided some condition on the parameter of the random walks is satisfied. The solutions of these equations are shown to be locally constant if their initial values are. Moreover, the heat-type equations can be identified with a system of ordinary differential equations. Conditions for the measure-valued branching stable random walks to possess the property of quasi-self-similarity are given, as well as a sufficient and necessary condition for these processes to be locally extinct. The latter result is consistent with the Euclidean case in the sense that the critical value for measure-valued branching stable processes to be locally extinct is the Hausdorff dimension of the image of the underlying processes divided by the dimension of the state space.

Absolute continuity of catalytic measure-valued branching processes

Classical super-Brownian motion (SBM) is known to take values in the space of absolutely continuous measures only if d=1. For d⩾2 its values are almost surely singular with respect to Lebesgue measure. This result has been generalized to more general motion laws and branching laws (yielding different critical dimensions) and also to catalytic SBM. In this paper we study the case of a catalytic measure-valued branching process in R d with a Feller process ξ as motion process, where the branching rate is given by a continuous additive functional of ξ, and where also the (critical) branching law may vary in space and time. We provide a simple sufficient condition for absolute continuity of the values of this process. This criterion is sharp for the classical cases. As a partial converse we also give a sufficient condition for singularity of the states.

Extinction of Superdiffusions and Semilinear Partial Differential Equations

A superdiffusion is a measure-valued branching process associated with a nonlinear operatorLu−ψ(u) whereLis a second order elliptic differential operator andψis a function from Rd×R+to R+. In the caseL1⩽0 (the so-called subcritical case), the expectation of the total mass does not increase and the mass vanishes at a finite time with probability 1. Most results on connections between the superdiffusion and differential equations involvingLu−ψ(u) were obtained for the subcritical case. In the present paper, we assume only thatL1 is a bounded function. In this more general setting, the probability of extinction can be smaller than 1, and we show that his happens if and only if there exists a strictly positive solution of the equationLu−ψ(u)=0 in Rd. We also establish a relation between the probability of extinction in a domainDand strictly positive solutions of equationLu−ψ(u)=0 inDthat are equal to 0 on the boundary ofD. We call such solutions special. For the equation (Δ+c)u=uα, whereΔis the Beltrami–Laplace operator on a complete Riemannian manifoldE, strictly positive solutions inEwere studied in connection with a geometrical problem: Which two functions represent scalar curvatures of two Riemannian metrics related by a conformal mapping (see [1] and references there)?

Absolute continuity of interacting measure-valued branching processes and its occupation-time processes

LetX t be the interaction measured-valued branchingα-symmetric stable process overR d (1< α ≤2) constructed by Meleard-Roelly[1]. Frist, it is shown thatX t is absolutely continuous with respect to the Lebesgue measure (onR) with a continuous density function which satisfies some SPDE. Second, it is proved that if the underlying process is a Brownian motion onR d(d≤3), the corresponding occupation-time processY t is also absolutely continuous with respect to the Lebesgue measure.

A LIMIT THEOREM FOR INTERACTING MEASURE-VALUED BRANCHING PROCESSES

It is well known that Fleming-Viot superprocesses can be obtained from the Dawson-Watanabe superprocesses by conditioning the latter to have constant total mass. The same question is investigated for measure-valued branching processes with interacting intensity independent of the geographical position. It is showed that a sequence of conditioned probability laws of this kind of interacting measure-valued branching processes also approximates to the probability law of Fleming-Viot superprocesses.

Immigration structures associated with Dawson-Watanabe superprocesses

The immigration structure associated with a measure-valued branching process may be described by a skew convolution semigroup. For the special type of measure-valued branching process, the Dawson-Watanabe superprocess, we show that a skew convolution semigroup corresponds uniquely to an infinitely divisible probability measure on the space of entrance laws for the underlying process. An immigration process associated with a Borel right superprocess does not always have a right continuous realization, but it can always be obtained by transformation from a Borel right one in an enlarged state space.