Conditional Log-Laplace Functionals of Immigration Superprocesses with Dependent Spatial Motion

Ergodicities and exponential ergodicities of Dawson-Watanabe type processes

Under natural assumptions, we prove the ergodicities and exponential ergodicities in Wasserstein and total variation distances of Dawson--Watanabe superprocesses without or with immigration. The strong Feller property in the total variation distance is derived as a by-product. The key of the approach is a set of estimates for the variations of the transition probabilities. The estimates in Wasserstein distance are derived from an upper bound of the kernels induced by the first moment of the superprocess. Those in total variation distance are based on a comparison of the cumulant semigroup of t…

Skeletal stochastic differential equations for superprocesses

It is well understood that a supercritical superprocess is equal in law to a discrete Markov branching process whose genealogy is dressed in a Poissonian way with immigration which initiates subcritial superprocesses. The Markov branching process corresponds to the genealogical description of prolific individuals, that is individuals who produce eternal genealogical lines of decent, and is often referred to as the skeleton or backbone of the original superprocess. The Poissonian dressing along the skeleton may be considered to be the remaining non-prolific genealogical mass in the superprocess…

Pathwise nonuniqueness for the SPDEs of some super-Brownian motions with immigration

We prove pathwise nonuniqueness in the stochastic partial differential equations (SPDEs) for some one-dimensional super-Brownian motions with immigration. In contrast to a closely related case investigated by Mueller, Mytnik and Perkins [Ann. Probab. 42 (2014) 2032-2112], the solutions of the present SPDEs are assumed to be nonnegative and have very different properties including uniqueness in law. In proving possible separation of solutions, we derive delicate properties of certain correlated approximating solutions, which is based on a novel coupling method called continuous decomposition. I…

Limit theorems for a class of critical superprocesses with stable branching

We consider a critical superprocess $\{X;\mathbf P_μ\}$ with general spatial motion and spatially dependent stable branching mechanism with lowest stable index $γ_0 > 1$. We first show that, under some conditions, $\mathbf P_μ(\|X_t\|\neq 0)$ converges to $0$ as $t\to \infty$ and is regularly varying with index $(γ_0-1)^{-1}$. Then we show that, for a large class of non-negative testing functions $f$, the distribution of $\{X_t(f);\mathbf P_μ(\cdot|\|X_t\|\neq 0)\}$, after appropriate rescaling, converges weakly to a positive random variable $\mathbf z^{(γ_0-1)}$ with Laplace transform $E[e^{-…

A control problem with fuel constraint and Dawson-Watanabe superprocesses

We solve a class of control problems with fuel constraint by means of the log-Laplace transforms of $J$-functionals of Dawson-Watanabe superprocesses. This solution is related to the superprocess solution of quasilinear parabolic PDEs with singular terminal condition. For the probabilistic verification proof, we develop sharp bounds on the blow-up behavior of log-Laplace functionals of $J$-functionals, which might be of independent interest.

Continuous-state branching processes with spectrally positive migration

Continuous-state branching processes (CSBPs) with immigration (CBIs), stopped on hitting zero, are generalized by allowing the process governing immigration to be any Lévy process without negative jumps. Unlike the CBIs, these newly introduced processes do not appear to satisfy any natural affine property on the level of the Laplace transforms of the semigroups. Basic properties are noted. Explicit formulae (on neighborhoods of infinity) for the Laplace transforms of the first passage times downwards and of the explosion time are derived.

On the hitting times of continuous-state branching processes with immigration

We study the two-dimensional joint distribution of the first hitting time of a constant level by a continuous-state branching process with immigration and their primitive stopped at this time. We show an explicit expression of its Laplace transform. Using this formula, we study the polarity of zero and provide a necessary and sufficient criterion for transience or recurrence. We follow the approach of Shiga, T. (1990) [A recurrence criterion for Markov processes of Ornstein-Uhlenbeck type. Probability Theory and Related Fields, 85(4), 425-447], by finding some $λ$-invariant functions for the g…

Some harmonic functions for killed Markov branching processes with immigration and culling

For a continuous-time Bienaymé-Galton-Watson process, $X$, with immigration and culling, $0$ as an absorbing state, call $X^q$ the process that results from killing $X$ at rate $q\in (0,\infty)$, followed by stopping it on extinction or explosion. Then an explicit identification of the relevant harmonic functions of $X^q$ allows to determine the Laplace transforms (at argument $q$) of the first passage times downwards and of the explosion time for $X$. Strictly speaking, this is accomplished only when the killing rate $q$ is sufficiently large (but always when the branching mechanism is not su…

Law of the exponential functional of a new family of one-sided Levy processes via self-similar continuous state branching processes with immigration and the Wright hypergeometric functions

We first introduce and derive some basic properties of a two-parameters family of one-sided Levy processes. Their Laplace exponents are given in terms of the Pochhammer symbol. This family includes, in a limit case, the family of Brownian motion with drifts. Then, we proceed by computing the density of the law of the exponential functional associated to some elements of this family (and their dual) and some transformations of these elements. These densities are expressed in terms of the Wright hypergeometric functions. By means of probabilistic arguments, we derive some interesting properties …

Continuous-state branching processes with immigration

This work provides a brief introduction to continuous-state branching processes (CB-processes) and continuous-state branching processes with immigration (CBI-processes) accessible to graduate students with reasonable background in probability theory and stochastic processes. In particular, we give a quick development of the stochastic equations of the processes and some immediate applications. The proofs given here are more elementary than those appearing in the literature before. We have made them readable without requiring too much preliminary knowledge on branching processes and stochastic …

Long-time behavior for subcritical measure-valued branching processes with immigration

In 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 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 sup…

通过其总种群数对逻辑连续状态分支过程在非灭绝条件下的条件化

Conditioning the logistic continuous-state branching process on non-extinction via its total progeny

本文研究了逻辑连续状态分支过程在非灭绝条件下的条件化，通过要求总种群数超过任意大的指数随机变量，构建了与Doob h变换相关的条件过程，并刻画了其有限寿命、随机方程表示及从零起始的性质。