Large scale localization of a spatial version of Neveu's branching process
Let \(M_f\) be the cone of all finite measures on \(\mathbb{R}^d\) equipped with the topology of weak convergence, \(D(\mathbb{R}_+, M_f)\) the Skorokhod space of all \(M_f\)-valued càdlàg paths, and \((X_t)\) the time-homogeneous Markov process in \(D(\mathbb{R}_+, M_f)\) characterized by its Laplace transition functional \(\mathbb{E}(\exp\{-X_t(\varphi)\}|X_0= \mu)= \exp\{-\mu(u_t[\varphi])\}\), \(t\geq 0\), \(\varphi= \mathbb{R}^d\to \mathbb{R}_+\) continuous with a positive infimum and a finite limit as \(|x|\uparrow\infty\), \(\mu\in M_f\), \(u_t= u_t[\varphi]\) the (unique) mild solution of the function-valued Cauchy problem \(du_t/dt= -(-\Delta)^{\alpha/2} u_t-\rho u_t\log u_t\), \(u_{0+}= \varphi\), \(0<\alpha\leq 2\), \(\rho> 0\) a fixed constant. Setting \(\widehat X^{(k)}(t)(B):= X_{kt}(k^{1/\alpha}B)/X_{kt}(1)\), \(t\geq 0\), \(B\) a Borel set, and letting \(k\to\infty\), it turns out that \((\widehat X^{(k)}(t_1),\dots,\widehat X^{(k)}(t_n))\to (\delta(\xi(t_1),\dots,\delta(\xi(t_n)))\) in law for every fixed collection of time points \(0\leq t_1<\cdots< t_n\), and if \(\alpha= 2\), \(\widehat X^{(k)}\to \delta(\xi)\) in law on \(D(\mathbb{R}_+, M_f)\), where \(\delta(x)\) is the point measure putting the total (unit) mass on \(x\), and \((\delta(t))\) is the \(\alpha\)-stable motion on \(\mathbb{R}^d\) starting at the origin: This is different from the ``usual'' types of behaviour of supercritical spatial branching processes.
scientific article