Convergence to equilibrium for density dependent Markov jump processes

Abstract

We investigate the convergence to (quasi--)equilibrium of a density dependent Markov chain in~${\mathbb{Z}}^d$, whose drift satisfies a system of ordinary differential equations having an attractive fixed point. For a sequence of such processes~${\mathbb{X}}^N$, indexed by a size parameter~$N$, the time taken until the distribution of~${\mathbb{X}}^N$, started in some given state, approaches its (quasi--)equilibrium distribution~$\ddot{I}{€}^N$ typically increases with~$N$. To first order, it corresponds to the time~$t_N$ at which the solution to the drift equations reaches a distance of~$\sqrt{N}$ from their fixed point. However, the length of the time interval over which the total variation distance between $\mathcal{L}({\mathbb{X}}^N(t))$ and its (quasi--)equilibrium distribution~$\ddot{I}{€}^N$ changes from being close to~$1$ to being close to zero is asymptotically of smaller order than~$t_N$. In this sense, the chains exhibit `cut--off', and we are able to prove that the cut-off window is of (optimal) constant size.

45 pages; accepted for publication in EJP