  This work continues and substantially extends our recent work on switching
diffusions with the switching processes that depend on the past states and that
take values in a countable state space. That is, the discrete components of the
two-component processes take values in a countably infinite set and its
switching rates at current time depend on the current value of the continuous
component. This paper focuses on recurrence, positive recurrence, and weak
stabilization of such systems. In particular, the paper aims to providing more
verifiable conditions on recurrence and positive recurrence and related issues.
Assuming that the system is linearizable, it provides feasible conditions
focusing on the coefficients of the systems for positive recurrence. Then
linear feedback controls for weak stabilization are considered. Some
illustrative examples are also given.
