马氏过程在鞍不动点附近的收敛性
Convergence of Markov processes near saddle fixed points
摘要：
考虑二维马氏过程序列$(X_t^N)_{t\geq0}$，其流动极限是$\dot{x}_t=b(x_t)$形式的常微分方程的稳定解，其中对于某些$λ,μ>0$和$τ(x)=O(|x|^2)$，$b(x)={\pmatrix{-μ0 0 λ}}x+τ(x)$.这里，对过程进行索引，使得$X_t^N$的波动方差与N成反比。最简单的例子来自于OK畜栏枪战模型，该模型由Williams和McIlroy [Bull. London Math.Soc.30（1998）166- 170]建立，并由Kingman [Bull. London Math.Soc.31（1999）601- 606]进行了研究。这些过程在$\log N$阶时表现出最有趣的行为，因此有必要建立一个在大时间内有效的流体极限。我们发现这个极限是内在随机的，并得到它的分布。利用这一点，可以推导出这些过程通过原点的直线上的点的缩放限制，以及它们可以达到的距原点的最小距离。给出适当缩放的N的幂是令人惊讶的。例如，如果T是$X_t^N$第一次命中y=x或y=-x之一的时间，则对于某个零均值高斯随机变量Z，\[N^{μ/{(2(λ+μ))}}|X_T^N|\Rightarrow |Z|^{μ/{(λ+μ)}},\]。
英文摘要：
We consider sequences $(X_t^N)_{t\geq0}$ of Markov processes in two dimensions whose fluid limit is a stable solution of an ordinary differential equation of the form $\dot{x}_t=b(x_t)$, where $b(x)={\pmatrix{-μ0 0 λ}}x+τ(x)$ for some $λ,μ>0$ and $τ(x)=O(|x|^2)$. Here the processes are indexed so that the variance of the fluctuations of $X_t^N$ is inversely proportional to N. The simplest example arises from the OK Corral gunfight model which was formulated by Williams and McIlroy [Bull. London Math. Soc. 30 (1998) 166--170] and studied by Kingman [Bull. London Math. Soc. 31 (1999) 601--606]. These processes exhibit their most interesting behavior at times of order $\log N$ so it is necessary to establish a fluid limit that is valid for large times. We find that this limit is inherently random and obtain its distribution. Using this, it is possible to derive scaling limits for the points where these processes hit straight lines through the origin, and the minimum distance from the origin that they can attain. The power of N that gives the appropriate scaling is surprising. For example if T is the time that $X_t^N$ first hits one of the lines y=x or y=-x, then \[N^{μ/{(2(λ+μ))}}|X_T^N|\Rightarrow |Z|^{μ/{(λ+μ)}},\] for some zero mean Gaussian random variable Z.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
