Dawson-Watanabe超过程的局部条件作用
Local conditioning in Dawson-Watanabe superprocesses
摘要：
考虑局部有限的Dawson-Watanabe超过程$ξ=(ξ_t)$在具有$d\geq2$的$\mathsf{R}^d$中.我们的主要结果包括一些递归公式的时刻措施的$ξ$，连接到均匀布朗树，布朗蛇表示棕榈措施，连续性的条件矩密度，导致对偶强连续版本的多元棕榈分布，和本地近似$ξ_t$的一个固定集群$\tildeη$具有良好的连续性和标度特性。这一切都导致了一个渐近描述的条件分布$ξ_t$为一个固定的$t>0$，给定$ξ_t$收费的$\varepsilon$ -邻域的一些点$x_1,\ldots,x_n\in \mathsf{R}^d$。在极限为$\varepsilon\to0$时，对这些集合的限制是条件独立的，并由伪随机测度$\tildeξ$或$\tildeη$给出，而对外部的贡献由$ξ_t$在$x_1,\ldots,x_n$的Palm分布给出。我们的证据是基于Cox集群表示的历史过程，并涉及一些微妙的时刻密度估计。
英文摘要：
Consider a locally finite Dawson-Watanabe superprocess $ξ=(ξ_t)$ in $\mathsf{R}^d$ with $d\geq2$. Our main results include some recursive formulas for the moment measures of $ξ$, with connections to the uniform Brownian tree, a Brownian snake representation of Palm measures, continuity properties of conditional moment densities, leading by duality to strongly continuous versions of the multivariate Palm distributions, and a local approximation of $ξ_t$ by a stationary cluster $\tildeη$ with nice continuity and scaling properties. This all leads up to an asymptotic description of the conditional distribution of $ξ_t$ for a fixed $t>0$, given that $ξ_t$ charges the $\varepsilon$-neighborhoods of some points $x_1,\ldots,x_n\in \mathsf{R}^d$. In the limit as $\varepsilon\to0$, the restrictions to those sets are conditionally independent and given by the pseudo-random measures $\tildeξ$ or $\tildeη$, whereas the contribution to the exterior is given by the Palm distribution of $ξ_t$ at $x_1,\ldots,x_n$. Our proofs are based on the Cox cluster representations of the historical process and involve some delicate estimates of moment densities.
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