Pitman-Rogers in Continuous Time and Arbitrary Space
Moving to Continuous Time
The technicalities of the Pitman-Rogers theorem are easiest to handle for discrete time Markov chains taking values in a discrete state space. But generalizing the result is not particularly difficult.
To understand what should happen to the intertwining relation $\mathbf{\Lambda} P = Q \mathbf{\Lambda}$ in the continuous time setting, just think of continuous time as the “many step” limit of a discrete time system. In discrete time the intertwining relation plus some simple algebra gives $$ \mathbf{\Lambda} P^{n+1} = Q^{n+1} \mathbf{\Lambda} $$ for all integers $n \geq 0$, since $$ \mathbf{\Lambda} P^{n+1} = \mathbf{\Lambda} P P^n = Q \mathbf{\Lambda} P^n, $$ and this formula can be iterated $n$ more times. This relation can be transported to the continuous time setting. In continuous time the families ${ P^n : n \geq 0 }$ and ${Q^n : n \geq 0 }$ get replaced by Markov semigroups, i.e. families ${ P_t : t \geq 0 }$ and ${ Q_t : t \geq 0 }$ that satisfy $P_0 = Q_0 = I$ and $$ P_{t+s} = P_t P_s, \quad Q_{t+s} = Q_t Q_s. $$ The $P_t$ semigroup acts on functions $f : S \to \mathbb{R}$ via $$ (P_t f)(x) = \int_S f(y) \mathbb{P}_x(X_t \in dy), $$ while the $Q_t$ semigroup function acts on functions $g : S' \to \mathbb{R}$ in a similar way. With these semigroups in hand it is reasonable to expect that the Pitman-Rogers intertwining relations gets replaced by $$ \mathbf{\Lambda} P_t = Q_t \mathbf{\Lambda} \quad \textrm{ for all } t \geq 0. $$