General Boundary Conditions for Denumerable Markov Processes

Our problem is to describe all Markov transition functions in a denumerable state space $E$ satisfying the condition
$$ \frac{dp(t,x,y)}{dt}\bigg|_{t=0}=a(x,y)\quad(x\in E)\eqno(a) $$
with a given matrix $a(x,y)$. This problem is solved under the following additional restriction on the matrix $a(x,y)$: for any $\lambda>0$ the equation
$$ \sum_{y\in E}a(x,y)f(g)=\lambda f(x)\quad(x\in E) $$
has only a finite number of linearly independent solutions $f$. We introduce a special set of characteristics for each transition function subject to condition (a). In the case of birth and death process this set coincides with the natural set of constants and measures characterising the behaviour of trajectories near the boundary points. In the general case we describe some properties of our characteristics and establish one-to-one correspondence between all sets with these properties and all transition functions satisfying condition (a).