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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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