  This work focuses on recurrence and ergodicity of switching diffusions
consisting of continuous and discrete components, in which the discrete
component takes values in a countably infinite set and the rates of switching
at current time depend on the value of the continuous component over an
interval including certain past history. Sufficient conditions for recurrence
and ergodicity are given. Moreover, the relationship between systems of partial
differential equations and recurrence when the switching is past-independent is
established under suitable conditions.
