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Under natural conditions, we prove exponential ergodicity in the L 1 -Wasserstein distance of two-type continuous-state branching processes in Lévy random environments with immigration. Furthermore, we express precisely the parameters of the exponent. The coupling method and the conditioned branching property play an important role in the approach. Using the tool of superprocesses, ergodicity in total variation distance is also proved. [ABSTRACT FROM AUTHOR]

We consider on M(ℝd) (the set of all finite measures on ℝd) the evolution equation associated with the nonlinear operator F↦ΔF′+∑k⩾1bkFkF \mapsto \Delta F' + \sum\nolimits_{k \geqslant 1} b_k F^k , where F′ is the variational derivative of F and we show that it has a solution represented by means of the distribution of the d-dimensional Brownian motion and the non-local branching process on the finite configurations of M(ℝd), induced by the sequence (bk)k⩾1 of positive numbers such that ∑k⩾1bk⩽1\sum\nolimits_{k \geqslant 1} b_k \leqslant 1. It turns out that the representation also holds with the same branching process for the solution to the equation obtained replacing the Laplace operator by the generator of a Markov process on ℝd instead of the d-dimensional Brownian motion; more general, we can take the generator of a right Markov process on a Lusin topological space. We first investigate continuous flows driving branching processes. We show that if the branching mechanism of a superprocess is independent of the spatial variable, then the superprocess is obtained by introducing the branching in the time evolution of the right continuous flow on measures, canonically induced by a right continuous flow as spatial motion. A corresponding result holds for non-local branching processes on the set of all finite configurations of the state space of the spatial motion.

A system of mutually interacting superprocesses with migration is constructed as the limit of a sequence of branching particle systems arising from population models. The uniqueness in law of the superprocesses is established using the pathwise uniqueness of a system of stochastic partial differential equations, which is satisfied by the corresponding system of distribution function-valued processes. [ABSTRACT FROM AUTHOR]

In this paper, we provide a pathwise spine decomposition for superprocesses with both local and non-local branching mechanisms under a martingale change of measure. This result complements earlier results established for superprocesses with purely local branching mechanisms and for multitype superprocesses. As an application of this decomposition, we obtain necessary/sufficient conditions for the limit of the fundamental martingale to be non-degenerate. In particular, we obtain extinction properties of superprocesses with non-local branching mechanisms as well as a KestenStigum L log L theorem for the fundamental martingale. [ABSTRACT FROM AUTHOR]

Suppose that X is a subcritical superprocess. Under some asymptotic conditions on the mean semigroup of X , we prove the Yaglom limit of X exists and identify all quasi-stationary distributions of X. [ABSTRACT FROM AUTHOR]

We generalize the notion of the submartingale property and Doob's inequality. Furthermore, we show how the latter leads to new inequalities for several stochastic processes: certain time series, Lévy processes, random walks, processes with independent increments, branching processes and continuous state branching processes, branching diffusions and superdiffusions, as well as some Markov processes, including geometric Brownian motion. [ABSTRACT FROM AUTHOR]

In this paper we establish a weak and a strong law of large numbers for supercritical superprocesses with general non-local branching mechanisms. Our results complement earlier results obtained for superprocesses with only local branching. Several interesting examples are developed, including multitype continuous-state branching processes, multitype superdiffusions and superprocesses with discontinuous spatial motions and non-decomposable branching mechanisms. [ABSTRACT FROM AUTHOR]

In previous work of D. Turaev, A. Winter and the author, the Law of Large Numbers for the local mass of certain superdiffusions was proved under an ergodicity assumption. In this paper we go beyond ergodicity, that is we consider cases when the scaling for the expectation of the local mass is not purely exponential. Inter alia, we prove the analog of theWatanabe–Biggins LLN for super-Brownian motion.

The purpose of this article is to give a rather thorough understanding of the compact support property for measure-valued processes corresponding to semi-linear equations of the form u(t) = Lu + beta u - alpha u(p) in R-d x (0, infinity), p is an element of (1, 2]; u(x, 0) = f(x) in R-d; u (x, t) >= 0 in R-d x [0, infinity). In particular, we shall investigate how the interplay between the underlying motion (the diffusion process corresponding to L) and the branching affects the compact support property. In [J. Englander, R. Pinsky, On the construction and support properties of measure-valued diffusions on D subset of R-d with spatially dependent branching, Ann. Probab. 27 (1999) 684-730], the compact support property was shown to be equivalent to a certain analytic criterion concerning uniqueness of the Cauchy problem for the semi-linear parabolic equation related to the measured valued process. In a subsequent paper [J. Englander, R. Pinsky, Uniqueness/nonuniqueness for nonnegative solutions of second-order parabolic equations of the form u(t) = Lu + Vu - gamma u(p) in R-n, J. Differential Equations 192 (2003) 396-428], this analytic property was investigated purely from the point of view of partial differential equations. Some of the results obtained in this latter paper yield interesting results concerning the compact support property. In this paper, the results from [J. Englander, R. Pinsky, Uniqueness/nonuniqueness, for nonnegative solutions of second-order parabolic equations of the form u(t) = Lu + Vu - gamma u(p) in R-n, J. Differential Equations 192 (2003) 396-428] that are relevant to the compact support property are presented, sometimes with extensions. These results are interwoven with new results and some informal heuristics. Taken together, they yield a rather comprehensive picture of the compact support property. Inter alia, we show that the concept of a measure-valued process hitting a point can be investigated via the compact support property, and suggest an alternate proof of a result concerning the hitting of points by super-Brownian motion. (c) 2005 Elsevier SAS. All rights reserved.

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. [ABSTRACT FROM AUTHOR]

We construct a class of superprocesses with interactive branching, immigration mechanisms, and spatial motion. It arises as the limit of a sequence of interacting branching particle systems with immigration, which generalizes a result of Méléard and Roelly (1993) established for a superprocess with interactive spatial motion. The uniqueness in law of the superprocess is established under certain conditions using the pathwise uniqueness of an SPDE satisfied by its corresponding distribution function process. This generalizes the recent work of Mytnik and Xiong (2015), where the result for a super-Brownian motion with interactive immigration mechanisms was obtained. An extended Yamada–Watanabe argument is used in the proving of pathwise uniqueness. [ABSTRACT FROM AUTHOR]

Consider a supercritical superprocess X = { X, t ⩾ 0} on a locally compact separable metric space ( E, m). Suppose that the spatial motion of X is a Hunt process satisfying certain conditions and that the branching mechanism is of the form where $$a \in B_b (E)$$, $$b \in B_b^ + (E)$$, and n is a kernel from E to (0,+∞) satisfying sup∫ y n( x, d y) < +∞. Put $$T_t f(x) = \mathbb{P}_{\delta _x } \left\langle {f,X_t } \right\rangle$$. Suppose that the semigroup { T; t ⩾ 0} is compact. Let λ be the eigenvalue of the (possibly non-symmetric) generator L of { T} that has the largest real part among all the eigenvalues of L, which is known to be real-valued. Let ϕ and $$\hat \varphi _0$$ be the eigenfunctions of L and $$\hat L$$ (the dual of L) associated with λ, respectively. Assume λ > 0. Under some conditions on the spatial motion and the ϕ-transform of the semigroup { T}, we prove that for a large class of suitable functions f, for any finite initial measure µ on E with compact support, where W is the martingale limit defined by $$W_\infty : = \lim _{t \to + \infty } e^{ - \lambda _0 t} \left\langle {\varphi _0 ,X_t } \right\rangle$$. Moreover, the exceptional set in the above limit does not depend on the initial measure µ and the function f. [ABSTRACT FROM AUTHOR]

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 (2014). [ABSTRACT FROM AUTHOR]

Suppose that X={ X: t≥0} is a supercritical super Ornstein-Uhlenbeck process, that is, a superprocess with an Ornstein-Uhlenbeck process on $\mathbb{R}^{d}$ corresponding to $L=\frac{1}{2}\sigma^{2}\Delta-b x\cdot\nabla$ as its underlying spatial motion and with branching mechanism ψ( λ)=− αλ+ βλ+∫( e−1+ λx) n( dx), where α=− ψ′(0+)>0, β≥0, and n is a measure on (0,∞) such that ∫ x n( dx)<+∞. Let $\mathbb{P} _{\mu}$ be the law of X with initial measure μ. Then the process W= e∥ X∥ is a positive $\mathbb{P} _{\mu}$-martingale. Therefore there is W such that W→ W, $\mathbb{P} _{\mu}$-a.s. as t→∞. In this paper we establish some spatial central limit theorems for X. Let $\mathcal{P}$ denote the function class For each $f\in\mathcal{P}$ we define an integer γ( f) in term of the spectral decomposition of f. In the small branching rate case α<2 γ( f) b, we prove that there is constant $\sigma_{f}^{2}\in (0,\infty)$ such that, conditioned on no-extinction, where W has the same distribution as W conditioned on no-extinction and $G_{1}(f)\sim \mathcal{N}(0,\sigma_{f}^{2})$. Moreover, W and G( f) are independent. In the critical rate case α=2 γ( f) b, we prove that there is constant $\rho_{f}^{2}\in (0,\infty)$ such that, conditioned on no-extinction, where W has the same distribution as W conditioned on no-extinction and $G_{2}(f)\sim \mathcal{N}(0, \rho_{f}^{2})$. Moreover W and G( f) are independent. We also establish two central limit theorems in the large branching rate case α>2 γ( f) b. Our central limit theorems in the small and critical branching rate cases sharpen the corresponding results in the recent preprint of Miłoś in that our limit normal random variables are non-degenerate. Our central limit theorems in the large branching rate case have no counterparts in the recent preprint of Miłoś. The main ideas for proving the central limit theorems are inspired by the arguments in K. Athreya's 3 papers on central limit theorems for continuous time multi-type branching processes published in the late 1960's and early 1970's. [ABSTRACT FROM AUTHOR]

We construct two kinds of stochastic flows of discrete Galton-Watson branching processes. Some scaling limit theorems for the flows are proved, which lead to local and nonlocal branching superprocesses over the positive half line. [ABSTRACT FROM AUTHOR]

In this note, we prove that any Fleming-Viot process on a Polish space is strongly Markovian provided so is the mutation process. [ABSTRACT FROM AUTHOR]

A class of interacting superprocesses arising from branching particle systems with continuous spatial motions, called superprocesses with dependent spatial motion (SDSMs), has been introduced and studied by Wang and by Dawson, Li and Wang. In this paper, we extend the model to allow discontinuous spatial motions. Under Lipschitz condition for coefficients, we show that under a proper rescaling, branching particle systems with jump-diffusion underlying motions in a random medium converge to a measure-valued process, called stable SDSM. We further characterize this stable SDSM as a unique solution of a well-posed martingale problem. To prove the uniqueness of the martingale problem, we establish the C2+γ-regularity for the transition semigroup of a class of jump-diffusion processes, which may be of independent interest. [ABSTRACT FROM AUTHOR]

For a one-dimensional superprocess in random environment, a nonlinear SPDE was derived by Dawson et al. (Ann Inst Henri Poincaré Probab Stat 36(2):167-180, ) for its density process. The time-space joint continuity of the density process was left as an open problem. In this paper we give an affirmative answer to this problem. [ABSTRACT FROM AUTHOR]

Abstract: A superprocess limit for an interacting birth–death particle system modeling a population with trait and physical age-structures is established. Traits of newborn offspring are inherited from the parents except when mutations occur, while ages are set to zero. Because of interactions between individuals, standard approaches based on the Laplace transform do not hold. We use a martingale problem approach and a separation of the slow (trait) and fast (age) scales. While the trait marginals converge in a pathwise sense to a superprocess, the age distributions, on another time scale, average to equilibria that depend on traits. The convergence of the whole process depending on trait and age, only holds for finite-dimensional time-marginals. We apply our results to the study of examples illustrating different cases of trade-off between competition and senescence. [Copyright &y& Elsevier]

We develop potential-theoretical methods in the construction of measure-valued branching processes. We complete results of P. J. Fitzsimmons and E. B. Dynkin on the construction, regularity and other properties of the superprocess associated with a given right process and a branching mechanism. [ABSTRACT FROM AUTHOR]

Central limit theorems of the occupation time of a superprocess over a stochastic flow are proved. For the critical and higher dimensions d≥4, the limits are Gaussian variables. For d=3, the limit is conditional Gaussian. When the stochastic flow disappears, the results degenerate to those for the ordinary super-Brownian motion. [ABSTRACT FROM AUTHOR]

Abstract: We establish Donnelly–Kurtz-type particle representations for a class of superprocesses with dependent spatial motions, and for a sequence of such superprocesses we prove convergence of the finite-dimensional distributions given convergence of the motion processes. As special cases, we construct a superprocess with coalescing spatial motion (SCSM) and a superprocess with dependent spatial motion (SDSM), where the underlying motion processes are one-dimensional coalescing and dependent Brownian motions, respectively. Under suitable conditions on the functions governing the interactions, we show convergence in distribution in of a sequence of SDSMs to an SCSM. [ABSTRACT FROM AUTHOR]

Abstract: We study the long time limiting behavior of the occupation time of the superprocess over a stochastic flow introduced by Skoulakis and Adler (2001) . The ergodic theorems for dimensions and are established. The proofs depend heavily on a characterization of the conditional log-Laplace equation of the occupation time process. [Copyright &y& Elsevier]

For a superprocess under a stochastic flow in one dimension, we prove that it has a density with respect to the Lebesgue measure. A stochastic partial differential equation is derived for the density. The regularity of the solution is then proved by using Krylov's L[subp]-theory for linear SPDE. [ABSTRACT FROM AUTHOR]

Abstract: We construct a class of discontinuous superprocesses with dependent spatial motion and general branching mechanism. The process arises as the weak limit of critical interacting–branching particle systems where the spatial motions of the particles are not independent. The main work is to solve the martingale problem. When we turn to the uniqueness of the process, we generalize the localization method introduced by [Daniel W. Stroock, Diffusion processes associated with Lévy generators, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 32 (1975) 209–244] to the measure-valued context. As for existence, we use particle system approximation and a perturbation method. This work generalizes the model introduced in [Donald A. Dawson, Zenghu Li, Hao Wang, Superprocesses with dependent spatial motion and general branching densities, Electron. J. Probab. 6 (25) (2001) 33 pp (electronic)] where a quadratic branching mechanism was considered. We also investigate some properties of the process. [Copyright &y& Elsevier]

We identify a class of conditional entrance laws for superprocesses with dependent spatial motion (SDSM). Those entrance laws are used to characterize some conditional excursion laws. As an application of the results, we give a sample path decomposition of the SDSM and that of a related immigration superprocess. The main tool used here is the conditional log-Laplace functional technique that handles the difficulty of the loss of the multiplicative property due to the interactions in the spatial motions. [ABSTRACT FROM AUTHOR]

A purely atomic immigration superprocess with dependent spatial motion in the space of tempered measures is constructed as the unique strong solution of a stochastic integral equation driven by Poisson processes based on the excursion law of a Feller branching diffusion, which generalizes the work of Dawson and Li [3]. As an application of the stochastic equation, it is proved that the superprocess possesses a local time which is Hölder continuous of order α for every α < 1/2. We establish two scaling limit theorems for the immigration superprocess, from which we derive scaling limits for the corresponding local time. [ABSTRACT FROM AUTHOR]

Abstract: A fissioning organism may purge unrepairable damage by bequeathing it preferentially to one of its daughters. Using the mathematical formalism of superprocesses, we propose a flexible class of analytically tractable models that allow quite general effects of damage on death rates and splitting rates and similarly general damage segregation mechanisms. We show that, in a suitable regime, the effects of randomness in damage segregation at fissioning are indistinguishable from those of randomness in the mechanism of damage accumulation during the organism''s lifetime. Moreover, the optimal population growth is achieved for a particular finite, non-zero level of combined randomness from these two sources. In particular, when damage accumulates deterministically, optimal population growth is achieved by a moderately unequal division of damage between the daughters, while too little or too much division is sub-optimal. Connections are drawn both to recent experimental results on inheritance of damage in protozoans, and to theories of aging and resource division between siblings. [Copyright &y& Elsevier]

Abstract: We define the Burgers superprocess to be the solution of the stochastic partial differential equation where , , and is space-time white noise. Taking gives the classic Burgers equation, an important, non-linear, partial differential equation. Taking 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. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]

Abstract: The purpose of this article is to give a rather thorough understanding of the compact support property for measure-valued processes corresponding to semi-linear equations of the form In particular, we shall investigate how the interplay between the underlying motion (the diffusion process corresponding to L) and the branching affects the compact support property. In [J. Engländer, R. Pinsky, On the construction and support properties of measure-valued diffusions on with spatially dependent branching, Ann. Probab. 27 (1999) 684–730], the compact support property was shown to be equivalent to a certain analytic criterion concerning uniqueness of the Cauchy problem for the semi-linear parabolic equation related to the measured valued process. In a subsequent paper [J. Englan̈der, R. Pinsky, Uniqueness/nonuniqueness for nonnegative solutions of second-order parabolic equations of the form in , J. Differential Equations 192 (2003) 396–428], this analytic property was investigated purely from the point of view of partial differential equations. Some of the results obtained in this latter paper yield interesting results concerning the compact support property. In this paper, the results from [J. Englan̈der, R. Pinsky, Uniqueness/nonuniqueness for nonnegative solutions of second-order parabolic equations of the form in , J. Differential Equations 192 (2003) 396–428] that are relevant to the compact support property are presented, sometimes with extensions. These results are interwoven with new results and some informal heuristics. Taken together, they yield a rather comprehensive picture of the compact support property. Inter alia, we show that the concept of a measure-valued process hitting a point can be investigated via the compact support property, and suggest an alternate proof of a result concerning the hitting of points by super-Brownian motion. [Copyright &y& Elsevier]

In this paper, we give a unified construction for superprocesses with dependent spatial motion constructed by Dawson, Li, Wang and superprocesses of stochastic flows constructed by Ma and Xiang. Furthermore, we also give some examples and rescaled limits of the new class of superprocesses. [ABSTRACT FROM AUTHOR]

Abstract: Pinsky [R.G. Pinsky, Transience, recurrence and local extinction properties of the support for supercritical finite measure-valued diffusions, Ann. Probab. 24 (1) 237–267] proved that the finite mass superdiffusion X corresponding to the semilinear operator exhibits local extinction if and only if , where is the generalized principal eigenvalue of on . For the case when , it has been shown in Engländer and Turaev [J. Engländer, D. Turaev, A scaling limit theorem for a class of superdiffusions, Ann. Probab. 30 (2) 683–722] that in law the superdiffusion locally behaves like times a non-negative non-degenerate random variable, provided that the operator satisfies a certain spectral condition (‘product-criticality’), and that α and are ‘not too large’. In this article we will prove that the convergence in law used in the formulation in [J. Engländer, D. Turaev, A scaling limit theorem for a class of superdiffusions, Ann. Probab. 30 (2) 683–722] can actually be replaced by convergence in probability. Furthermore, instead of we will consider a general Euclidean domain . As far as the proof of our main theorem is concerned, the heavy analytic method of [J. Engländer, D. Turaev, A scaling limit theorem for a class of superdiffusions, Ann. Probab. 30 (2) 683–722] is replaced by a different, simpler and more probabilistic one. We introduce a space–time weighted superprocess (H-transformed superprocess) and use it in the proof along with some elementary probabilistic arguments. [Copyright &y& Elsevier]

A class of finite measure-valued càdlàg superprocesses X with Neveu''s (1992) continuous-state 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 prove convergence on path space as β↓0. The log-Laplace equation related to X has the locally non-Lipschitz function ulogu as non-linear term (instead of u1+β in the case of X(β)). It can nevertheless be shown to be well-posed. X has infinite expectation, is immortal in all finite times, propagates mass instantaneously everywhere in space, and has locally countably infinite biodiversity. [Copyright &y& Elsevier]

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). [ABSTRACT FROM AUTHOR]