Ergodic theorems for linear growth processes with diffusion
Let \({\mathbb{E}}\) be the set of all \(\eta\) : \({\mathbb{Z}}^ d\to {\mathbb{Z}}_+\). The pregenerator of the process studied in the paper is as follows: \[ \Omega f(\eta)=\sum_{x\in {\mathbb{Z}}^ d}\{(\lambda_ 1\eta (x)+\lambda_ 3)[f[\eta +e_ x)-f(x)]+\lambda_ 2\eta (x)[f(\eta -e_ x)-f(x)]\} \] \[ +\sum_{x,y\in {\mathbb{Z}}^ d}\eta (x)p(x,y)[f(\eta -e_ x+e_ y)-f(\eta)] \] where \(\lambda_ 1,\lambda_ 2,\lambda_ 3\geq 0\), \(e_ x\in {\mathbb{E}}\) is the configuration valued one at x and zero elsewhere, (p(x,y)) is a transition probability matrix on \({\mathbb{Z}}^ d\). Suppose that \[ \sup_{x}\sum_{y}p(y,x)=K<\infty. \] One of the main results in the paper for the case \(\lambda_ 3=0\) claims that the process is ergodic if and only if \(\lambda_ 1-\lambda_ 2+K-1<0\) and it is not if \(\lambda_ 1-\lambda_ 2>0\) plus \(\inf_{t>0}\sum_{u}p(t,u,x)>0\) for some x, where \[ p(t,x,u)=\sum^{\infty}_{n=0}(t^ n/n!)p^{(n)}(u,x). \] Moreover, the authors study the construction of the invariant measures in the particular case that \(\lambda_ 3=0\) and \(\lambda_ 1=\lambda_ 2>0\). For the case that \(\lambda_ 3>0\), the ergodic and non-ergodic regions are also described.
scientific article; zbMATH DE number 4117623