A cyclically catalytic super-Brownian motion
type: Item
instance of: scholarly article
title: A cyclically catalytic super-Brownian motion
zbMATH Open document ID: 1017.60099
author: Klaus Fleischmann
author: Jie Xiong
published in: The Annals of Probability
publication date: 2003-05-06T00:00:00Z
review text: The authors try to extend the mutually catalytic model to \(K\) types \(A^0, \dots, A^{K-1}\) of materials, and restrict in particular to a cyclic situation \(A^k + A^{k+1}\) \(\rightarrow\) \(A^k\), \(k \in {\mathcal K}\), where \({\mathcal K} =\) \(\{ 0, 1, \dots, K-1\}\) denotes the cyclic group of size \(K \geq 2\). The following stochastic equation is considered: \[ dX_t^k(a) = \frac{\sigma^2}{2} \Delta X_t^k(a)dt + \sqrt{ \gamma^k X_t^k(a) X_t^{k+1}(a)}d W_t^k(a), \tag{1} \] for \(t > 0\), \((k,a) \in {\mathcal K} \times {\mathbb R}\). Here \(\sigma > 0\) is the diffusion constant, \(\gamma^k > 0\) are the interaction rates, and \(dW\) denotes a standard white noise on \({\mathbb R}_+ \times {\mathcal K} \times {\mathbb R}\). The quantity \(X_t^k(a)\) can be interpreted as the density of mass of type \(k\) at time \(t\) at site \(a\). Intuitively, the subpopulation \(X^k\) of type \(k\) evolves as a super-Brownian motion (SBM) in \({\mathbb R}\), but with branching rate \(\gamma^k X_t^{k+1}(a)\) changing with \(t\) and \(a\). In other words, the subpopulation \(X^{k+1}\) serves as a catalyst for the branching of \(X^k\) for each \(k\). This true interaction of types destroys the basic independence assumption in branching theory, so that this model \(X\) is not a superprocess in its standard definition. The first purpose of the paper is to establish that a weak solution \(X\) to (1) exists which is a strong Markov process. Theorem (a). To each \(K \geq 2\), \(\sigma > 0\) and vector \(\gamma = ( \gamma^k)\), \(k \in {\mathcal K}\), with \(\gamma^k > 0\), there exists a cyclically catalytic super-Brownian motion \(( X, P_x\), \(x \in C_{\text{tem}}^+)\) in \({\mathcal K} \times {\mathbb R}\) with diffusion constant \(\sigma\) and interaction rate \(\gamma\). \noindent In addition, this \(X\) has finite moments of all orders. In fact, it is shown that, for fixed \(c, T, q > 0\), and \(\mu, \lambda \in {\mathbb R}\) with \(q \mu < \lambda\), \[ \sup \left\{ P_x \s…
reviewed by: Isamu Dôku
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60J80
zbMATH DE Number: 1906002
zbMATH Keywords: catalyst
zbMATH Keywords: cyclic reaction
zbMATH Keywords: martingale problem
zbMATH Keywords: global segregation of neighboring types
zbMATH Keywords: finite time survival
zbMATH Keywords: extinction
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1008956694
OpenAlex ID: W2083519015
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DOI: 10.1214/AOP/1008956694
description: scientific article; zbMATH DE number 1906002