连续介质自由跳跃动力学
Free Jump Dynamics in Continuum
摘要：
描述了无限大跳跃点粒子系统在$\mathbb{R}^d$中的演化。系统的状态是粒子组态空间上的概率测度。在初始状态$μ_0$具有如下各阶相关函数的条件下：（a）$k_{μ_0}^{(n)} \in L^\infty ((\mathbb{R}^d)^n)$（本质有界）;（b）$\|k_{μ_0}^{(n)}\|_{ L^\infty ((\mathbb{R}^d)^n)} \leq C^n$，$n\in \mathbb{N}$（亚泊松），演化$μ_0 \mapsto μ_t$，$t>0$在本质有界的亚泊松函数空间中被得到为连续可微映射$k_{μ_0} \mapsto k_t$，$k_t =(k_t^{(n)})_{n\in \mathbb{N}}$.特别是，它被证明$k_t$解决了相应的发展方程，并为每个$t>0$，它是一个独特的状态$μ_t$的相关函数。
英文摘要：
The evolution is described of an infinite system of hopping point particles in $\mathbb{R}^d$. The states of the system are probability measures on the space of configurations of particles. Under the condition that the initial state $μ_0$ has correlation functions of all orders which are: (a) $k_{μ_0}^{(n)} \in L^\infty ((\mathbb{R}^d)^n)$ (essentially bounded); (b) $\|k_{μ_0}^{(n)}\|_{ L^\infty ((\mathbb{R}^d)^n)} \leq C^n$, $n\in \mathbb{N}$ (sub-Poissonian), the evolution $μ_0 \mapsto μ_t$, $t>0$, is obtained as a continuously differentiable map $k_{μ_0} \mapsto k_t$, $k_t =(k_t^{(n)})_{n\in \mathbb{N}}$, in the space of essentially bounded sub-Poissonian functions. In particular, it is proved that $k_t$ solves the corresponding evolution equation, and that for each $t>0$ it is the correlation function of a unique state $μ_t$.
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