测度值扩散方程的紧支撑性质
The compact support property for measure-valued diffusions
摘要：
本文的目的是对对应于\[& u_t=Lu+βu-αu^p \text{in} R^d\times (0,\infty), p\in(1,2]; &u(x,0)=f(x) \text{in} R^d; &u(x,t)\ge0 \text{in} R^d\times[0,\infty). \]形式的半线性方程的测度值扩散过程的紧支集性质给出一个相当透彻的理解。特别地，我们将研究基本运动（对应于$L$的扩散过程）和分支之间的相互作用如何影响紧支集性质。在\cite{EP99}中，证明了紧支撑性质等价于与测值扩散有关的半线性抛物型方程Cauchy问题唯一性的一个解析判据.在随后的论文\cite{EP03}中，纯粹从偏微分方程的角度研究了这种分析性质。在这后一份文件中得到的一些结果产生有趣的结果关于紧支持属性。在本文中，从\cite{EP03}的结果是相关的紧支撑性质，有时与扩展。这些结果与新的结果和一些非正式的理论是交织在一起的。综合起来，它们给出了紧凑支撑特性的相当全面的描述。\it除其他外\rm，我们表明，一个测度值扩散\it击中\rm点的概念可以通过紧支持性质进行研究，并建议一个关于超布朗运动击中点的结果的另一种证明。
英文摘要：
The purpose of this article is to give a rather thorough understanding of the compact support property for measure-valued diffusion processes corresponding to semi-linear equations of the form \[& u_t=Lu+βu-αu^p \text{in} R^d\times (0,\infty), p\in(1,2]; &u(x,0)=f(x) \text{in} R^d; &u(x,t)\ge0 \text{in} R^d\times[0,\infty). \] In particular, we shall investigate how the interplay between the underlying motion (the diffusion process corresponding to $L$) and the branching affects the compact support property. In \cite{EP99}, the compact support property was shown to be equivalent to a certain analytic criterion concerning uniqueness of the Cauchy problem for the semilinear parabolic equation related to the measured valued diffusion. In a subsequent paper \cite{EP03}, this analytic property was investigated purely from the point of view of partial differential equations. Some of the results obtained in this latter paper yield interesting results concerning the compact support property. In this paper, the results from \cite{EP03} that are relevant to the compact support property are presented, sometimes with extensions. These results are interwoven with new results and some informal heuristics. Taken together, they yield a fairly comprehensive picture of the compact support property. \it Inter alia\rm, we show that the concept of a measure-valued diffusion \it hitting\rm a point can be investigated via the compact support property, and suggest an alternate proof of a result concerning the hitting of points by super-Brownian motion.
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