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It is well known that the dynamics of a subpopulation of individuals of a rare type in a Wright-Fisher diffusion can be approximated by a Feller branching process. Here we establish an analogue of that result for a spatially distributed population whose dynamics are described by a spatial Lambda-Fleming-Viot process (SLFV). The subpopulation of rare individuals is then approximated by a superBrownian motion. This result mirrors Cox et al. (2000), where it is shown that when suitably rescaled, sparse voter models converge to superBrownian motion. We also prove the somewhat more surprising result, that by choosing the dynamics of the SLFV appropriately we can recover superBrownian motion with stable branching in an analogous way. This is a spatial analogue of (a special case of) results of Bertoin and Le Gall (2006), who show that the generalised Fleming-Viot process that is dual to the beta-coalescent, when suitably rescaled, converges to a continuous state branching process with stable branching mechanism., 43 pages, 0 figures

We decompose the genealogy of a general superprocess with spatially dependent branching mechanism with respect to the last individual alive (Williams decomposition). This is a generalization of the main result of Delmas and H\'{e}nard [Electron. J. Probab.,18,1-43,2013] where only superprocesses with spatially dependent quadratic branching mechanism were considered. As an application of the Williams decomposition, we prove that, for some superprocesses, the normalized total mass will converge to a point mass at its extinction time. This generalizes a result of Tribe [Ann. Probab.,20,286-311,1992] in the sense that our branching mechanism is more general.

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 {\it et al.} (2001) and Wang (1997, 1998).

We construct a flow of continuous time and discrete state branching processes. Some scaling limit theorems for the flow are proved, which lead to the path-valued branching processes and nonlocal branching superprocesses over the positive half line studied in Li (2012). Mathematics Subject Classification (2010): Primary 60J68, 60J80; secondary 60G57

Let $X=\{X_{t},t\ge0;\mathbb{P}_{\mu}\}$ be a critical superprocess starting from a finite measure $\mu$. Under some conditions, we first prove that $\lim_{t\to\infty}t{ \mathbb{P}}_{\mu}(\Vert X_{t}\Vert \ne0)=\nu^{-1}\langle\phi_{0},\mu\rangle$, where $\phi_{0}$ is the eigenfunction corresponding to the first eigenvalue of the infinitesimal generator $L$ of the mean semigroup of $X$, and $\nu$ is a positive constant. Then we show that, for a large class of functions $f$, conditioning on $\Vert X_{t}\Vert \ne0$, $t^{-1}\langle f,X_{t}\rangle$ converges in distribution to $\langle f,\psi_{0}\rangle_{m}W$, where $W$ is an exponential random variable, and $\psi_{0}$ is the eigenfunction corresponding to the first eigenvalue of the dual of $L$. Finally, if $\langle f,\psi_{0}\rangle_{m}=0$, we prove that, conditioning on $\Vert X_{t}\Vert \ne0$, $(t^{-1}\langle\phi_{0},X_{t}\rangle,t^{-1/2}\langle f,X_{t}\rangle )$ converges in distribution to $(W,G(f)\sqrt{W})$, where $G(f)\sim\mathcal{N}(0,\sigma_{f}^{2})$ is a normal random variable, and $W$ and $G(f)$ are independent.

We consider an approximating sequence of interacting population models with branching, mutation and competition. Each individual is characterized by its trait and the traits of its ancestors. Birth- and death-events happen at exponential times. Traits are hereditarily transmitted unless mutation occurs. The present model is an extension of the model used in [M\'el\'eard and Tran, EJP, 2012], where for large populations with small individual biomasses and under additional assumptions, the diffusive limit is shown to converge to a nonlinear historical superprocess. The main goal of the present article is to verify a compact containment condition in the more general setup of Polish trait-spaces and general mutation kernels that allow for a dependence on the parent's trait. As a by-product, a result on the paths of individuals is obtained. An application to evolving genealogies on marked metric measure spaces is mentioned where genealogical distance, counted in terms of the number of births without mutation, can be regarded as a trait. Because of the use of exponential times in the modeling of birth- and death-events the analysis of the modulus of continuity of the trait-history of a particle plays a major role in obtaining appropriate bounds., Comment: 13 pages. Shortened version with shortened and streamlined proofs

The totally disconnectedness of support for super Brownian motion in high dimensions is well known. In this paper, we prove that similar results also hold for $\Lambda$-Fleming-Viot process with Brownian spatial motion provided that the associated $\Lambda$-coalescent does not come down from infinity fast enough. Our proof is another application of the lookdown particle representation for $\Lambda$-Fleming-Viot process. We also discuss the disjointness of independent $\Lambda$-Fleming-Viot supports and ranges in high dimensions. The disconnectedness of the $\Lambda$-Fleming-Viot support remains open in certain low dimensions.

The model under consideration is a catalytic branching model constructed in Dawson and Fleischmann (1997), where the catalysts themselves undergo a spatial branching mechanism. The key result is a convergence theorem in dimension d = 3 towards a limit with full intensity (persistence), which, in a sense, is comparable with the situation for the “classical” continuous super-Brownian motion. As by-products, strong laws of large numbers are derived for the Brownian collision local time controlling the branching of reactants, and for the catalytic occupation time process. Also, the catalytic occupation measures are shown to be absolutely continuous with respect to Lebesgue measure. © 1997 Elsevier Science B.V.

A super-Brownian motion X in ℝ with "hyperbolic" branching rate ρ2 (b) = 1/b2, b ∈ ℝ, is constructed, which symbolically could be described by the formal stochastic equation dXt = ½ ΔXt dt + √2ρ2XtdWt, t ≥ 0, (1) (with a space-time white noise W). If the finite starting measure X0 does not have mass at b = 0, then this superprocess X will never hit the catalytic center: There is Brownian stopping time r strictly smaller than the hitting time of 0 such that Dynkin's stopped measures Xr vanishes except a.s.

Soit une superdiffusion $X$ sur $\mathbb{R}^{d}$ correspondant a l’operateur semi-lineaire $\mathcal{A}(u)=Lu+\beta u-ku^{2}$, ou $L$ est lui-meme un operateur eliptique du second ordre, $\beta(\cdot)$ est dans la classe de Kato, et $k(\cdot)\ge0$ est borne sur les compacts de $\mathbb{R}^{d}$ et est positif sur un ensemble de mesure de Lebesgue positive. L’objectif principal de cet article est de completer les resultats obtenus dans (Ann. Probab. 32 (2004) 78–99), dans le sens suivant. Soit $\lambda_{\infty}$ la borne $L^{\infty}$ de croissance du semigroupe correspondant a l’operateur $L+\beta$ de type Schrodinger. Si $\lambda_{\infty}\neq0$, nous prouvons alors que – dans un certain sens – le taux exponentiel de croissance/decroissance de la masse totale $\|X_{t}\|$, est $\lambda_{\infty}$. Nous decrivons egalement le comportement limite de $\exp(-\lambda_{\infty}t)\|X_{t}\|$, quand $t\to\infty$, sous cette meme hypothese. Ces resultats sont a comparer avec ceux obtenus dans (Ann. Probab. 32 (2004) 78–99), ou il est demontre que la valeur propre principale generalisee $\lambda_{2}$ de l’operateur donne le taux de croissance locale quand elle est positive et qu’il y a extinction locale quand ce n’est pas le cas. Il est aise de montrer que $\lambda_{\infty}\ge\lambda_{2}$, et nous discutons les cas $\lambda_{\infty}>\lambda_{2}$ et $\lambda_{\infty}=\lambda_{2}$. Quand $\lambda_{\infty}=0$, et sous certaines conditions portant sur $\beta$, nous obtenons une condition necessaire et suffisante pour que la superdiffusion $X$ s’eteigne faiblement. Nous montrons que l’intensite de branchement $k$ affecte l’extinction faible; alors qu’il est connu que $k$ n’affecte pas l’extinction faible locale. (Celle-ci dependant uniquement du signe de $\lambda_{2}$ et est equivalente a l’extinction locale.)

Consider a locally finite Dawson-Watanabe superprocess $\xi=(\xi_t)$ in $\mathsf{R}^d$ with $d\geq2$. Our main results include some recursive formulas for the moment measures of $\xi$, with connections to the uniform Brownian tree, a Brownian snake representation of Palm measures, continuity properties of conditional moment densities, leading by duality to strongly continuous versions of the multivariate Palm distributions, and a local approximation of $\xi_t$ by a stationary cluster $\tilde{\eta}$ with nice continuity and scaling properties. This all leads up to an asymptotic description of the conditional distribution of $\xi_t$ for a fixed $t>0$, given that $\xi_t$ charges the $\varepsilon$-neighborhoods of some points $x_1,\ldots,x_n\in \mathsf{R}^d$. In the limit as $\varepsilon\to0$, the restrictions to those sets are conditionally independent and given by the pseudo-random measures $\tilde{\xi}$ or $\tilde{\eta}$, whereas the contribution to the exterior is given by the Palm distribution of $\xi_t$ at $x_1,\ldots,x_n$. Our proofs are based on the Cox cluster representations of the historical process and involve some delicate estimates of moment densities., Comment: Published in at http://dx.doi.org/10.1214/11-AOP702 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We show that density functions of a $(\alpha,1,\beta)$-superprocesses are almost sure multifractal for $\alpha>\beta+1$, $\beta\in(0,1)$ and calculate the corresponding spectrum of singularities., Comment: Published at http://dx.doi.org/10.1214/14-AOP951 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

A new approach is provided to the (critical continuous) super-Brownian motion X in R with a single point-catalyst δc as branching rate. We start from a superprocess U with constant branching rate and spatial motion given by the stable subordinator with index 1/2. We prove that the total occupation time measure ∫0∞ ds Us of U is distributed as the occupation density measure λc of X at the catalyst c. This result is a superprocess analogue of the classical fact that the set of zeros of a linear Brownian motion is the range of a stable subordinator with index 1/2. We then show that the value Xt of the process X at time t is determined from the measure λc by an explicit representation formula. On a heuristic level, this formula says that a mass λc(ds) of ''particles'' leaves the catalyst at time s and then evolves according to the Itô measure of Brownian excursions. This representation formula has important applications. First of all, with probability one, the density field x of X satisfies the heat equation outside of c with the noisy boundary condition at c given by the singularly continuous random measure λc. In particular, x is C∞ outside the catalyst. This property is in sharp contrast to the constant branching rate case. Another consequence is that the total mass Xt(R) is always strictly positive but dies out in probability as t → ∞. As a final application a new derivation of the singularity of the measure λc is provided.

Continuous super-Brownian motions in two and higher dimensions are known to have singular measure states. However, by weakening the branching mechanism in an irregular way they can be forced to have absolutely continuous states. The sufficient conditions we impose are identified in a couple of examples with irregularities in only one coordinate. This includes the case of branching restricted to some densely situated ensemble of hyperplanes.

This paper examines the existence of the self-intersection local time for a superprocess over a stochastic flow in dimensions $d\leq3$, which through constructive methods, results in a Tanaka-like representation. The superprocess over a stochastic flow is a superprocess with dependent spatial motion, and thus Dynkin's proof of existence, which requires multiplicity of the log-Laplace functional, no longer applies. Skoulakis and Adler's method of calculating moments is extended to higher moments, from which existence follows., Comment: Published in at http://dx.doi.org/10.1214/11-AOP653 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We are interested in the evolving genealogy of a birth and death process with trait structure and ecological interactions. Traits are hereditarily transmitted from a parent to its offspring unless a mutation occurs. The dynamics may depend on the trait of the ancestors and on its past and allows interactions between individuals through their lineages. We define an interacting historical particle process describing the genealogies of the living individuals; it takes values in the space of point measures on an infinite dimensional c\`adl\`ag path space. This individual-based process can be approximated by a nonlinear historical superprocess, under the assumptions of large populations, small individuals and allometric demographies. Because of the interactions, the branching property fails and we use martingale problems and fine couplings between our population and independent branching particles. Our convergence theorem is illustrated by two examples of current interest in biology. The first one relates the biodiversity history of a population and its phylogeny, while the second treats a spatial model with competition between individuals through their past trajectories., Comment: 31 pages

A family of continuous-state branching processes with immigration are constructed as the solution flow of a stochastic equation system driven by time-space noises. The family can be regarded as an inhomogeneous increasing path-valued branching process with immigration. Two nonlocal branching immigration superprocesses can be defined from the flow. We identify explicitly the branching and immigration mechanisms of those processes. The results provide new perspectives into the tree-valued Markov processes of Aldous and Pitman [Ann. Inst. Henri Poincar\'{e} Probab. Stat. 34 (1998) 637-686] and Abraham and Delmas [Ann. Probab. 40 (2012) 1167-1211]., Comment: Published in at http://dx.doi.org/10.1214/12-AOP759 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We present a genealogy for superprocesses with a non-homogeneous quadratic branching mechanism, relying on a weighted version of the superprocess and a Girsanov theorem. We then decompose this genealogy with respect to the last individual alive (William's decomposition). Letting the extinction time tend to infinity, we get the Q-process by looking at the superprocess from the root, and define another process by looking from the top. Examples including the multitype Feller diff usion and the superdiffusion are provided., 43 pages

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., Comment: Published in at http://dx.doi.org/10.1214/10-AOP574 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

For a superprocess under a stochastic flow, we prove that it has a density with respect to the Lebesgue measure for d=1 and is singular for d>1. For d=1, a stochastic partial differential equation is derived for the density. The regularity of the solution is then proved by using Krylov's L_p-theory for linear SPDE. A snake representation for this superprocess is established. As applications of this representation, we prove the compact support property for general d and singularity of the process when d>1.

In a previous paper of Winter and the author the Law of Large Numbers for the local mass of certain superdiffusions was proved under a spectral theoretical assumption, which is equivalent to the ergodicity (positive recurrence) of the motion component of an $H$-transformed (or weighted) superprocess. In fact the assumption is also equivalent to the property that the scaling for the expectation of the local mass is pure exponential. In this paper we go beyond ergodicity, that is we consider cases when the scaling is not purely exponential. Inter alia, we prove the analog of the Watanabe-Biggins Law of Large Numbers for super-Brownian motion (SBM). We will also prove another Law of Large Numbers for a bounded set moving with subcritical speed, provided the variance term decays sufficiently fast., 20 pages. See also http://www.pstat.ucsb.edu/faculty/englander/publications.html

The $(d,\alpha,\beta,\gamma)$-branching particle system consists of particles moving in $\mathbb{R}^d$ according to a symmetric $\alpha$-stable L\'evy process $(0 d)$. In some cases $H_T\equiv 1$ and in others $H_T\to\infty$ as $T\to\infty$ (high density systems). The limit processes are quite different for Lebesgue and for finite measures. Therefore the question arises of what kinds of limits can be obtained for Poisson intensity measures that are intermediate between Lebesgue measure and finite measures. In this paper the measures $\mu_\gamma, \gamma\in (0,d]$, are used for investigating this question. Occupation time fluctuation limits are obtained which interpolate in some way between the two previous extreme cases. The limit processes depend on different arrangements of the parameters $d,\alpha,\beta,\gamma$. There are two thresholds for the dimension $d$. The first one, $d=\alpha/\beta+\gamma$, determines the need for high density or not in order to obtain non-trivial limits, and its relation with a.s. local extinction of the system is discussed. The second one, $d=[\alpha(2+\beta)-\gamma\vee \alpha]/\beta$\ (if $\gamma < d$), interpolates between the two extreme cases, and it is a critical dimension which separates different qualitative behaviors of the limit processes, in particular long-range dependence in ``low'' dimensions, and independent increments in ``high'' dimensions. In low dimensions the temporal part of the limit process is a new self-similar stable process which has two different long-range dependence regimes depending on relationships among the parameters. Related results for the corresponding $(d,\alpha,\beta,\gamma)$-superprocess are also given.

Let $\xi$ be a Dawson--Watanabe superprocess in $\mathbb{R}^d$ such that $\xi_t$ is a.s. locally finite for every $t\geq 0$. Then for $d\geq2$ and fixed $t>0$, the singular random measure $\xi_t$ can be a.s. approximated by suitably normalized restrictions of Lebesgue measure to the $\varepsilon$-neighborhoods of $\operatorname {supp}\xi_t$. When $d\geq3$, the local distributions of $\xi_t$ near a hitting point can be approximated in total variation by those of a stationary and self-similar pseudo-random measure $\tilde{\xi}$. By contrast, the corresponding distributions for $d=2$ are locally invariant. Further results include improvements of some classical extinction criteria and some limiting properties of hitting probabilities. Our main proofs are based on a detailed analysis of the historical structure of $\xi$., Comment: Published in at http://dx.doi.org/10.1214/07-AOP386 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We consider the behaviour of a continuous super-Brownian motion catalysed by a random medium with infinite overall density under the hydrodynamic scaling of mass, time, and space. We show that, in supercritical dimensions, the scaled process converges to a macroscopic heat flow, and the appropriately rescaled random fluctuations around this macroscopic flow are asymptotically bounded, in the sense of log-Laplace transforms, by generalised stable Ornstein-Uhlenbeck processes. The most interesting new effect we observe is the occurrence of an index-jump from a 'Gaussian' situation to stable fluctuations of index 1+gamma, where gamma is an index associated to the medium., 40 pages

We consider a discrete model that describes a locally regulated spatial population with mortality selection. This model was studied in parallel by Bolker and Pacala and Dieckmann, Law and Murrell. We first generalize this model by adding spatial dependence. Then we give a pathwise description in terms of Poisson point measures. We show that different normalizations may lead to different macroscopic approximations of this model. The first approximation is deterministic and gives a rigorous sense to the number density. The second approximation is a superprocess previously studied by Etheridge. Finally, we study in specific cases the long time behavior of the system and of its deterministic approximation., Published at http://dx.doi.org/10.1214/105051604000000882 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We study a nonlinear stochastic partial differential equation whose solution is the conditional log-Laplace functional of a superprocess in a random environment. We establish its existence and uniqueness by smoothing out the nonlinear term and making use of the particle system representation developed by Kurtz and Xiong [Stochastic Process. Appl. 83 (1999) 103-126]. We also derive the Wong-Zakai type approximation for this equation. As an application, we give a direct proof of the moment formulas of Skoulakis and Adler [Ann. Appl. Probab. 11 (2001) 488-543]., Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/009117904000000540

In a supercritical branching particle system, the trimmed tree consists of those particles which have descendants at all times. We develop this concept in the superprocess setting. For a class of continuous superprocesses with Feller underlying motion on compact spaces, we identify the trimmed tree, which turns out to be a binary splitting particle system with a new underlying motion that is a compensated h-transform of the old one. We show how trimmed trees may be estimated from above by embedded binary branching particle systems., Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/009117904000000090

A non-critical branching immigration superprocess with dependent spatial motion is constructed and characterized as the solution of a stochastic equation driven by a time-space white noise and an orthogonal martingale measure. A representation of its conditional log-Laplace functionals is established, which gives the uniqueness of the solution and hence its Markov property. Some properties of the superprocess including an ergodic theorem are also obtained.

A class of interacting superprocesses, called superprocesses with dependent spatial motion (SDSMs), has been introduced and characterized in Wang citeWang98 and Dawson et al. citeDLW01. In this paper, we give a construction or an excursion representation of the non-degenerate SDSM with immigration by making use of a Poisson system associated with the conditional excursion laws of the SDSM. As pointed out in Wang citeWang98, the multiplicative property or summable property is lost for SDSMs and immigration SDSMs. However, summable property is the foundation of excursion representation. This raises a sequence of technical difficulties. The main tool we used is the conditional log-Laplace functional technique that gives the conditional summability, the conditional excursion law, and the Poisson point process for the construction of the immigration SDSMs.

In the single point catalytic super-Brownian motion "particles" branch only if they meet the position of the single point catalyst. If the branching rate tends to zero, the model degenerates to the heat flow. We are concerned with large deviation probabilities related to this law of large numbers. To this aim the well-known explicit representation of the model by excursion densities is heavily used. The rate function is described by the Fenchel-Legendre transform of log-exponential moments described by a log-Laplace equation.

Let X be either the branching diffusion corresponding to the operator $Lu+\beta (u^2-u)$ on $D\subseteq $ $\mathbb{R}^{d}$ [where $\beta (x) \geq 0$ and $\beta\not\equiv 0$ is bounded from above] or the superprocess corresponding to the operator $Lu+\beta u -\alpha u^2$ on $D\subseteq $ $\mathbb{R}^{d}$ (with $\alpha>0$ and $\beta$ is bounded from above but no restriction on its sign). Let $\lambda _{c}$ denote the generalized principal eigenvalue for the operator $L+\beta $ on $D$. We prove the following dichotomy: either $\lambda _{c}\leq 0$ and X exhibits local extinction or $\lambda _{c}> 0$ and there is exponential growth of mass on compacts of D with rate $\lambda _{c}$. For superdiffusions, this completes the local extinction criterion obtained by Pinsky [Ann. Probab. 24 (1996) 237--267] and a recent result on the local growth of mass under a spectral assumption given by Engländer and Turaev [Ann. Probab. 30 (2002) 683--722]. The proofs in the above two papers are based on PDE techniques, however the proofs we offer are probabilistically conceptual. For the most part they are based on "spine'' decompositions or "immortal particle representations'' along with martingale convergence and the law of large numbers. Further they are generic in the sense that they work for both types of processes.

We study the limit of a superprocess controlled by a stochastic flow as $t\to\infty$. It is proved that when $d \le 2$, this process suffers long-time local extinction; when $d\ge 3$, it has a limit which is persistent. The stochastic log-Laplace equation conjectured by Skoulakis and Adler (2001) and studied by this author (2004) plays a key role in the proofs like the one played by the log-Laplace equation in deriving long-term behavior for usual super-Brownian motion.

We study the global properties of transition semigroups $(p_t^{\nu , \Psi , A})$ of $(A, \Psi )$-superprocesses over compact type spaces with possibly nonzero immigration $\nu$ in various function spaces. In particular, we compare the different rates of convergence of $(p_t^{\nu ,\Psi ,A})$ to equilibrium. Our analysis is based on an explicit formula for the Gateaux derivative of $p_t^{\nu ,\Psi , A} F$.

A uniqueness problem raised in 2001 for critical cyclically catalytic super-Brownian motions is solved in the simplified space-less case, that is, for cyclically catalytic branching diffusions X. More precisely, X is characterized as the unique strong solution of a singular stochastic equation.

Consider a system of particles which move in R^d according to a symmetric alpha-stable motion, have a lifetime distribution of finite mean, and branch with an offspring law of index 1+beta. In case of the critical dimension d=alpha/beta, 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+beta. Our result generalizes the case alpha=2 of Brownian particles of Klenke (1998), where pde methods had been used which are not available in the present setting.

For a continuous super-Brownian reactant X in Rd with general catalyst ρ a Brownian snake representation is derived for the part Xc of X in the catalyst region. This extends results of Dawson et al. (2002) and Klenke (2003) in that it allows the collision local time of an intrinsic reactant particle with the catalyst to have flat pieces, caused by catalyst-free regions.

Impressed by Neveu's (1992) continuous-state branching process we learned about from Bertoin and Le Gall (2000), a class of finite measure-valued càdlàg superprocesses X with Neveu's branching mechanism is constructed. To this end, we start from certain supercritical (α,d,β)-superprocesses X(β) with symmetric α-stable motion and (1+β)-branching and let β↓0. The log-Laplace equation related to X has the locally non-Lipschitz function 푢 log 푢 as non-linear term (instead of 푢1+β in the case of X(β)) and is thus interesting in its own. X has infinite expectations, is immortal in all finite times, propagates mass instantaneously everywhere in space, and has locally countably infinite biodiversity.

We consider the stochastic heat equation with a multiplicative colored noise term on the real space for dimensions greater or equal to 1. First, we prove convergence of a branching particle system in a random environment to this stochastic heat equation with linear noise coefficients. For this stochastic partial differential equation with more general non-Lipschitz noise coefficients we show convergence of associated lattice systems, which are infinite dimensional stochastic differential equations with correlated noise terms, provided that uniqueness of the limit is known. In the course of the proof, we establish existence and uniqueness of solutions to the lattice systems, as well as a new existence result for solutions to the stochastic heat equation. The latter are shown to be jointly continuous in time and space under some mild additional assumptions.