Convergence of Markov processes near saddle fixed points

Abstract

We consider sequences (X_t^N)_t≥0 of Markov processes in two dimensions whose fluid limit is a stable solution of an ordinary differential equation of the form ẋ_t=b(x_t), where $b(x)=\bigl(\begin{smallmatrix}-\mu\ 0 \cr 0\ \lambda\end{smallmatrix}\bigr)x+\tau(x)$ 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 logN 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 $$N^{μ/{(2(λ+μ))}}|X_T^N| ⇒ |Z|^{μ/{(λ+μ)}},$$ for some zero mean Gaussian random variable Z.