State variable or jump variable
I'm assuming pollution is a bad in consumer utility (as is typically seen) or otherwise relevant to some agent's payoffs -- if it's not, then $P_t$ is irrelevant and you might as well ignore it.
$P_t$ is a state variable: it's something which describes the history of the system, and is necessary to compute utility. See page 117, Definition 3.1 in these notes for a more detailed definition of state variables (&quot;utility function&quot; = &quot;contribution function&quot;), the section has some more interesting discussion as well.
If pollution appeared in utility and you were solving a planner's problem, I think there would be no confusion since you would need $P_t$ to calculate the decision rule. Where your problem gets a bit different is that the agents don't use it to compute their decision rules. Mechanically, that makes your calculations a bit simpler: optimize agents' objectives over potential values of $K_t$ only, since the decision rule is going to be &quot;flat&quot; over $P_t$.
That an eigenvalue is bigger than one suggests that pollution or capital in this economy will &quot;explode&quot;. But that doesn't bear on whether $P_t$ is a state variable or not.