On the rate of convergence of spatial birth-and-death processes
A spatial birth-and-death process is a continuous-time Markov chain X(t) on a state space \(\Omega =\cup^{\infty}_{0}\Omega_ n\), where the \(\Omega_ n\) are disjoint and \(\Omega_ 0\) is a singleton. When the process is at \(x\in \Omega_ n\) then at the next transition it can only move to a state in \(\Omega_{n+1}\) or \(\Omega_{n-1}\) (at rates \(\beta\) (x), \(\delta\) (x) respectively). Let \(Q_ t(x,F)=P(X(t)\in F | X(0)=x)\) denote the transition distribution, for appropriate measurable subsets F of \(\Omega\). The author studies sufficient conditions for the distribution to converge geometrically fast to the invariant distribution. These results follow from conditions which ensure that \[ \sup_{F}| \int Q_ t(x,F)\gamma (dx)-\int Q_ t(y,F)\kappa (dy)| \] converges to zero geometrically fast, for suitable initial distributions \(\gamma\) and \(\kappa\).
scientific article; zbMATH DE number 4151560
scientific article