Strong uniqueness for a class of singular SDEs for catalytic branching diffusions

Catalytic branching processes via spine techniques and renewal theory

In this article we contribute to the moment analysis of branching processes in catalytic media. The many-to-few lemma based on the spine technique is used to derive a system of (discrete space) partial differential equations for the number of particles in a variation of constants formulation. The long-time behaviour is then deduced from renewal theorems and induction.

Infinite rate mutually catalytic branching in infinitely many colonies. Construction, characterization and convergence

We construct a mutually catalytic branching process on a countable site space with infinite "branching rate". The finite rate mutually catalytic model, in which the rate of branching of one population at a site is proportional to the mass of the other population at that site, was introduced by Dawson and Perkins in [DP98]. We show that our model is the limit for a class of models and in particular for the Dawson-Perkins model as the rate of branching goes to infinity. Our process is characterized as the unique solution to a martingale problem. We also give a characterization of the process as …

On the maximal displacement of catalytic branching random walk

We study the distribution of the maximal displacement of particles positions for the whole time of the population existence in the model of critical and subcritical catalytic branching random walk on Z. In particular, we prove that in the case of simple symmetric random walk on Z, the distribution of the maximal displacement has "a heavy tail" decreasing as a function of the power 1/2 or 1, when the branching process is critical or subcritical, respectively. These statements describe new effects which do not arise in the corresponding investigations of the maximal displacement of critical and …

Infinite rate mutually catalytic branching in infinitely many colonies: The longtime behavior

Consider the infinite rate mutually catalytic branching process (IMUB) constructed in [Infinite rate mutually catalytic branching in infinitely many colonies. Construction, characterization and convergence (2008) Preprint] and [Ann. Probab. 38 (2010) 479-497]. For finite initial conditions, we show that only one type survives in the long run if the interaction kernel is recurrent. On the other hand, under a slightly stronger condition than transience, we show that both types can coexist.

The Spread of a Catalytic Branching Random Walk

We consider a random walk on $\Z$ that branches at the origin only. In the supercritical regime we establish a law of large number for the maximal position $M_n$. Then we determine all possible limiting law for the sequence $M_n -αn$ where $α$ is a deterministic constant.

Mutually Catalytic Branching Processes on the Lattice and Voter Processes with Strength of Opinion

Since the seminal work of Dawson and Perkins, mutually catalytic versions of super-processes have been studied frequently. In this article we combine two approaches extending their ideas: the approach of adding correlations to the driving noise of the system is combined with the approach of obtaining new processes by letting the branching rate tend to infinity. We introduce infinite rate symbiotic branching processes which surprisingly can be interpreted as generalized voter processes with additional strength of opinions. Since many of the arguments go along the lines of known proofs this arti…

Generalized Mehler Semigroups and Catalytic Branching Processes with Immigration

Skew convolution semigroups play an important role in the study of generalized Mehler semigroups and Ornstein-Uhlenbeck processes. We give a characterization for a general skew convolution semigroup on real separable Hilbert space whose characteristic functional is not necessarily differentiable at the initial time. A connection between this subject and catalytic branching superprocesses is established through fluctuation limits, providing a rich class of non-differentiable skew convolution semigroups. Path regularity of the corresponding generalized Ornstein-Uhlenbeck processes in different t…

Genealogy of catalytic branching models

We consider catalytic branching populations. They consist of a catalyst population evolving according to a critical binary branching process in continuous time with a constant branching rate and a reactant population with a branching rate proportional to the number of catalyst individuals alive. The reactant forms a process in random medium. We describe asymptotically the genealogy of catalytic branching populations coded as the induced forest of $\mathbb{R}$-trees using the many individuals--rapid branching continuum limit. The limiting continuum genealogical forests are then studied in detai…

催化分支对称稳定过程的传播速率

Spread rate of catalytic branching symmetric stable processes

本文研究分支对称 α-稳定过程的最大位移增长阶数，在分支率测度属于 Kato 类且具有紧支撑的条件下，证明最大位移呈指数增长，并确定其阶数由指标 α 和相应 Schrödinger 型算子的谱底决定。

Longtime Behavior for Mutually Catalytic Branching with Negative Correlations

In several examples, dualities for interacting diffusion and particle systems permit the study of the longtime behavior of solutions. A particularly difficult model in which many techniques collapse is a two-type model with mutually catalytic interaction introduced by Dawson/Perkins for which they proved under some assumptions a dichotomy between extinction and coexistence directly from the defining equations. In the present article we show how to prove a precise dichotomy for a related model with negatively correlated noises. The proof uses moment bounds on exit-times of correlated Brownian m…

具有相互催化相互作用的分支粒子系统

Branching Particle Systems with Mutually Catalytic Interactions

本文研究了具有相互催化相互作用的分支粒子系统，探讨了两种种群在长期行为中的共存与不共存现象，并通过重整化过程和对偶技术分析其极限行为。

Degenerate Stochastic Differential Equations for Catalytic Branching Networks

Uniqueness of the martingale problem corresponding to a degenerate SDE which models catalytic branching networks is proven. This work is an extension of a paper by Dawson and Perkins to arbitrary catalytic branching networks. As part of the proof estimates on the corresponding semigroup are found in terms of weighted Holder norms for arbitrary networks, which are proven to be equivalent to the semigroup norm for this generalized setting. ----- On prouve l'unicite d'un probleme de martingale correspondant a une EDS degeneree, qui apparait comme un modele de reseaux avec branchement catalytique.…

平均场分支SDE：混沌传播、尺度极限与相变

Mean-field branching SDEs: propagation of chaos, scaling limits and phase transitions

该研究针对带依赖律迁入的分支SDE及其平均场粒子近似，建立混沌传播界、两阶段尺度极限，给出逻辑斯蒂平均场扩散灭绝/非灭绝判据，明确弱相互作用是时间一致近似的必要条件。

Infinite rate mutually catalytic branching

Consider the mutually catalytic branching process with finite branching rate $γ$. We show that as $γ\to\infty$, this process converges in finite-dimensional distributions (in time) to a certain discontinuous process. We give descriptions of this process in terms of its semigroup in terms of the infinitesimal generator and as the solution of a martingale problem. We also give a strong construction in terms of a planar Brownian motion from which we infer a path property of the process. This is the first paper in a series or three, wherein we also construct an interacting version of this process …

Classification of catalytic branching processes and structure of the criticality set

We study a catalytic branching process (CBP) with any finite set of catalysts. This model describes a system of particles where the movement is governed by a Markov chain with arbitrary finite or countable state space and the branching may only occur at the points of catalysis. The results obtained generalize and strengthen those known in cases of CBP with a single catalyst and of branching random walk on d-dimensional integer lattice with a finite number of sources of particles generation. We propose to classify CBP with N catalysts as supercritical, critical or subcritical according to the v…

Spread of a Catalytic Branching Random Walk on a Multidimensional Lattice

For a supercritical catalytic branching random walk on Z^d (d is positive integer) with an arbitrary finite catalysts set we study the spread of particles population as time grows to infinity. Namely, we divide by t the position coordinates of each particle existing at time t and then let t tend to infinity. It is shown that in the limit there are a.s. no particles outside the closed convex surface in R^d which we call the propagation front and, under condition of infinite number of visits of the catalysts set, a.s. there exist particles on the propagation front. We also demonstrate that the p…

Subcritical catalytic branching random walk with finite or infinite variance of offspring number

Subcritical catalytic branching random walk on d-dimensional lattice is studied. New theorems concerning the asymptotic behavior of distributions of local particles numbers are established. To prove the results different approaches are used including the connection between fractional moments of random variables and fractional derivatives of their Laplace transforms. In the previous papers on this subject only supercritical and critical regimes were investigated assuming finiteness of the first moment of offspring number and finiteness of the variance of offspring number, respectively. In the p…

A Trotter-type approach to infinite rate mutually catalytic branching

Dawson and Perkins [Ann. Probab. 26 (1988) 1088--1138] constructed a stochastic model of an interacting two-type population indexed by a countable site space which locally undergoes a mutually catalytic branching mechanism. In Klenke and Mytnik [Preprint (2008), arXiv:0901.0623], it is shown that as the branching rate approaches infinity, the process converges to a process that is called the infinite rate mutually catalytic branching process (IMUB). It is most conveniently characterized as the solution of a certain martingale problem. While in the latter reference, a noise equation approach is…

Catalytic Branching Random Walk with Semi-exponential Increments

A catalytic branching random walk on a multidimensional lattice, with arbitrary finite number of catalysts, is studied in supercritical regime. The dynamics of spatial spread of the particles population is examined, upon normalization. The components of the vector random walk jump are assumed independent (or close to independent) and have semi-exponential distributions with, possibly, different parameters. A limit theorem on the almost sure normalized positions of the particles at the population ``front'' is established. Contrary to the case of the random walk increments with ``light'' distrib…

Strong and weak convergence of population size in supercritical catalytic branching process

A general model of catalytic branching process (CBP) with any finite number of catalysis centers in a discrete space is studied. More exactly, it is assumed that particles move in this space according to a specified Markov chain and they may produce offspring only in the presence of catalysts located at fixed points. The asymptotic (in time) behavior of the total number of particles as well as the local particles numbers is investigated. The problems of finding the global extinction probability and local extinction probability are solved. Necessary and sufficient conditions are established for…

Maximum of Catalytic Branching Random Walk with Regularly Varying Tails

For a continuous-time catalytic branching random walk (CBRW) on Z, with an arbitrary finite number of catalysts, we study the asymptotic behavior of position of the rightmost particle when time tends to infinity. The mild requirements include the regular variation of the jump distribution tail for underlying random walk and the well-known L log L condition for the offspring numbers. In our classification, given in the previous paper, the analysis refers to supercritical CBRW. The principle result demonstrates that, after a proper normalization, the maximum of CBRW converges in distribution to …

Degenerate stochastic differential equations arising from catalytic branching networks

We establish existence and uniqueness for the martingale problem associated with a system of degenerate SDE's representing a catalytic branching network. For example, in the hypercyclic case: $$dX_{t}^{(i)}=b_i(X_t)dt+\sqrt{2γ_{i}(X_{t}) X_{t}^{(i+1)}X_{t}^{(i)}}dB_{t}^{i}, X_t^{(i)}\ge 0, i=1,..., d,$$ where $X^{(d+1)}\equiv X^{(1)}$, existence and uniqueness is proved when $γ$ and $b$ are continuous on the positive orthant, $γ$ is strictly positive, and $b_i>0$ on $\{x_i=0\}$. The special case $d=2$, $b_i=θ_i-x_i$ is required in work of Dawson-Greven-den Hollander-Sun-Swart on mean fields li…

Fluctuations of propagation front in catalytic branching walk

We consider a supercritical catalytic branching random walk (CBRW) on a multidimensional lattice Z^d (d is positive integer). The main subject of study is the behavior of particles cloud in space and time. For CBRW on an integer line, Carmona and Hu (2014) examined the asymptotical behavior of the maximal coordinate M_n of the particles at time n. They proved that M_n/n converges to μalmost surely (on a set of local non-degeneracy of CBRW), as n tends to infinity, where μ>0 is a certain constant. Under additional assumption of a single catalyst in CBRW they also investigated the fluctuations o…

关于亚临界自催化分支布朗运动的研究

On the subcritical self-catalytic branching Brownian motions

本文研究了自催化分支布朗运动的亚临界情况，构造了允许无限初始粒子的模型，并刻画了其来自无穷的CDI性质及速率。

催化分支随机游走的局域化相变

A localisation phase transition for the catalytic branching random walk

该研究针对带原点催化剂的连续时间分支随机游走，证明了局域化与非局域化相变的存在，发现其转变阈值与仅原点分支的λ=0情形一致，且在局域化相证明了占据测度的收敛性，优于现有λ=0情形仅矩收敛的结果。

Branching diffusions, superdiffusions and random media

Spatial branching processes became increasingly popular in the past decades, not only because of their obvious connection to biology, but also because superprocesses are intimately related to nonlinear partial differential equations. Another hot topic in today's research in probability theory is `random media', including the now classical problems on `Brownian motion among obstacles' and the more recent `random walks in random environment' and `catalytic branching' models. These notes aim to give a gentle introduction into some topics in spatial branching processes and superprocesses in determ…

Skeletal stochastic differential equations for continuous-state branching process

It is well understood that a supercritical continuous-state branching process (CSBP) is equal in law to a discrete continuous-time Galton Watson process (the skeleton of prolific individuals) whose edges are dressed in a Poissonian way with immigration which initiates subcritial CSBPs (non-prolific mass). Equally well understood in the setting of CSBPs and super-processes is the notion of a spine or immortal particle dressed in a Poissonian way with immigration which initiates copies of the original CSBP, which emerges when conditioning the process to survive eternally. In this article, we rev…

Optimal Portfolio Liquidation in Target Zone Models and Catalytic Superprocesses

We study optimal buying and selling strategies in target zone models. In these models the price is modeled by a diffusion process which is reflected at one or more barriers. Such models arise for example when a currency exchange rate is kept above a certain threshold due to central bank intervention. We consider the optimal portfolio liquidation problem for an investor for whom prices are optimal at the barrier and who creates temporary price impact. This problem will be formulated as the minimization of a cost-risk functional over strategies that only trade when the price process is located a…

一类BSDEs的分支过程数值算法

A numerical algorithm for a class of BSDEs via branching process

将半线性抛物型PDE的分支过程算法推广到非马尔可夫情形，用于求解一类倒向随机微分方程（BSDEs），并通过路径依赖PDE的粘性解概念证明算法收敛性。