A scaling analysis of a cat and mouse Markov chain

If $(C_n)$ is a Markov chain on a discrete state space $\mathcal{S}$, a Markov chain $(C_n,M_n)$ on the product space $\mathcal{S}\times \mathcal{S}$, the cat and mouse Markov chain, is constructed. The first coordinate of this Markov chain behaves like the original Markov chain and the second component changes only when both coordinates are equal. The asymptotic properties of this Markov chain are investigated. A representation of its invariant measure is, in particular, obtained. When the state space is infinite it is shown that this Markov chain is in fact null recurrent if the initial Markov chain $(C_n)$ is positive recurrent and reversible. In this context, the scaling properties of the location of the second component, the mouse, are investigated in various situations: simple random walks in $\mathbb{Z}$ and ${\mathbb{Z}}^2$ reflected a simple random walk in $\mathbb{N}$ and also in a continuous time setting. For several of these processes, a time scaling with rapid growth gives an interesting asymptotic behavior related to limiting results for occupation times and rare events of Markov processes.