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Construction of immigration superprocesses with dependent spatial motion from one-dimensional excursions. (English) Zbl 1038.60082
It is shown that for each \(\mu\in M(\mathbb{R})\) (finite Borel measures on \(\mathbb{R}\)) a unique probability measure \(Q_\mu\) on \(C([0,\infty),M(\mathbb{R}))\) exists such that, for each \(\varphi\in C^2(\mathbb{R})\), under \(Q_\mu\), \[ M_t(\varphi)= \langle\varphi, w_t\rangle- \langle\varphi, \mu\rangle- {1\over 2}\rho(0) \int^t_0\langle\varphi'', w_s\rangle\,ds, \quad t\geq 0,\tag{1} \] is a continuous martingale with quadratic variation process \[ \langle M(\varphi)\rangle_t= \int^t_0 \langle\sigma\varphi^2, w_s\rangle\,ds+ \int^t_0 \int_{\mathbb{R}}\langle h(z-\cdot)\, \varphi', w_s\rangle^2\,dz\,ds \tag{2} \] (\(w_s\) is the coordinate process), where \[ \rho(x)= \int_{\mathbb{R}} h(y- x)h(y)\,dy,\quad x\in\mathbb{R}, \] and \(h\) is a continuously differentiable function on \(\mathbb{R}\) such that \(h\) and \(h'\) are square-integrable. The system \(\{Q_\mu,\,\mu\in M(\mathbb{R})\}\) represents a superprocess with dependent spatial motion (SDSM), where \(\rho(0)\) is the migration rate and \(\sigma\) is the branching rate. Whereas super-Brownian motion (which corresponds to independence of the spatial motion) starting from any \(\mu\in M(\mathbb{R})\) immediately enters the space of absolutely continuous measures, SDSM starting from any \(\mu\) immediately enters the space of atomic measures.
The objective of the paper is to construct a class of immigration diffusion processes related to the SDSM, which consists in replacing (1) by \[ M_t(\varphi)= \langle\varphi, w_t\rangle- \langle\varphi, \mu\rangle-{1\over 2}\rho(0) \int^t_0 \langle\varphi'', w_s\rangle\,ds- \int^t_0 \langle\varphi q(w_s,\cdot), m\rangle \,ds,\quad t\geq 0,\tag{3} \] where \(m\) is a nontrivial \(\sigma\)-finite Borel measure on \(\mathbb{R}\) and \(q\) is a Borel function on \(M(\mathbb{R})\times \mathbb{R}\) with certain regularity conditions. The martingale problem (1) and (3) represents an SDSM with interactive immigration determined by \(q(w_s,\cdot)\) and reference measure \(m\). A new method is required for the construction of this diffusion process. It is constructed as the pathwise unique solution of a stochastic integral equation carried by a stochastic flow and driven by a Poisson process of one-dimensional excursions. The stochastic integral equation provides information on the properties of the sample paths of the immigration diffusion, which are given in detail.

MSC:
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60G57 Random measures
60H20 Stochastic integral equations
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