Construction of immigration superprocesses with dependent spatial motion from one-dimensional excursions.

*(English)*Zbl 1038.60082It 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.

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.

Reviewer: Louis G. Gorostiza (Mexico City)

##### MSC:

60J80 | Branching processes (Galton-Watson, birth-and-death, etc.) |

60G57 | Random measures |

60H20 | Stochastic integral equations |