A cyclically catalytic super-Brownian motion
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 \(q \mu < \lambda\), \[ \sup \left\{ P_x \sup_{ T \geq t \geq 0} \sum_{ k \in {\mathcal K}} \langle (X_t^k)^q, \varphi_{\lambda} \rangle ; x \in C_{\text{tem}}^+, c \geq |x |_{- \mu} \right\} < \infty, \tag{2} \] where \(\varphi_{\lambda}(a) = e^{- \lambda |a |}\), \(a \in {\mathbb R}\) and \(\langle f, g \rangle\) is an integral of \(f \cdot g\) with respect to \(da\). Here \(C_{\text{tem}}^+\) denotes the totality of all positive members of \(C_{\text{tem}}\) \(:=\) \(\bigcap_{\lambda > 0}\) \(C_{- \lambda}({\mathcal K} \times {\mathbb R})\), and \(C_{- \lambda}({\mathcal K} \times {\mathbb R})\) is the set of all continuous functions \(f\) on \({\mathcal K} \times {\mathbb R}\) such that \[ |f |_{- \lambda} = \sup_{k \in {\mathcal K}, a \in {\mathbb R}} |f^k(a) |\cdot \varphi_{\lambda}(a) < \infty, \] satisfying that \(f^k(a) \cdot \varphi_{\lambda}(a)\) has a finite limit as \(|a |\nearrow \infty\) for each \(k\). The existence result for \(K = 2\) was proved by \textit{D. A. Dawson} and \textit{E. A. Perkins} [Ann.\ Probab. 26, No. 3, 1088-1138 (1998; Zbl 0938.60042)]. Besides the construction of \(X\), their next target is the survival-extinction behavior of cyclically catalytic SBMs in the case of finite populations. More precisely, the authors prove noncoexistence of neighboring types: Theorem (b). If a cyclically catalytic SBM \(X\) has a finite initial mass \(\|X_0 \|\), then for total mass processes \(\{ \|X_t^k \|\}_k\) we have \[ \lim_{t \rightarrow \infty} \|X_t^k \|\cdot \|X_t^{k+1} \|= 0, \quad \text{a.s.} \tag{3} \] for each \(k \in {\mathcal K}\). Consequently, for each pair of neighboring types, only one of them has the chance to survive in the limit. Moreover, on the supposition that the process \(X\) has the initial process \(X_0 \in C_{\text{tem}}^+\) satisfying \(\|X_0 \|< \infty\), it is proved: Theorem (c). Assume that \(\prod_k \|X_0^k \|> 0\), and that there is a \(T > 0\) such that \[ \max_{ k \in {\mathcal K}} \liminf_{|a |\nearrow \infty} \frac{ S_t [ X_0^k]^2(a) \cdot S_t[ X_0^{k+1} ]^2(a)}{[ S_t X_0^k(a)]^4} = 0, \quad t \geq T. \tag{4} \] Then \[ \prod_{k \in {\mathcal K}} \|X_t^k \|> 0 \quad \text{for} \text{all} t > 0, \quad \text{a.s.} \tag{5} \] \noindent Here \(\{ S_t, t \geq 0 \}\) denotes the heat flow semigroup for the diffusion operator \(\frac{\sigma^2}{2} \Delta\). In fact, the above suggests that \(X_0\) can be chosen in such a way that all types survive for all finite times. And besides, the authors also succeed in obtaining some finite time extinction result. Roughly speaking, given a type \(k\) and a positive time \(t\), the \(k\)th subpopulation \(X^k\) dies by time \(t\) with high probability, provided that its initial value \(X_0^k\) is sufficiently small. This work is closely related to and technically in part due to \textit{C. Mueller} and \textit{E. A. Perkins} [Ann.\ Inst. Henri Poincaré, Probab.\ Stat. 36, No. 3, 301-338 (2000; Zbl 0966.60060)] and \textit{T. Shiga} [Can. J. Math. 46, No. 2, 415-437 (1994; Zbl 0801.60050)] as well as to the first quoted paper. This paper is well motivated and carefully written in its spirit in connection with the first quoted paper, and provides mathematically rich materials, which the reader will find interesting and stimulating. For example, the uniqueness of cyclically catalytic SBM remains unsolved if \(K \geq 3\), continuous dependence on the initial data of constructed solutions to the martingale problem is also an unsolved problem, and the question whether the limit \(( \|X_{\infty}^0 \|, \dots, \|X_{\infty}^{K-1} \|)\) is nondegenerate or not for \(K \geq 3\) remains open, etc.
scientific article; zbMATH DE number 1906002
scientific article