Good morning, I apologize in advance if my question sounds too basic but after some research I was unable to come up with satisfactory answers to my doubts.
I am currently studying a model which needs a slight modification of the notion of Piecewise Deterministic Markov Processes (PDMP). In short I am given a compact smooth manifold $M$ and a finite set $E$, a collection of smooth vector fields $(F^i)_{i\in E}$ on $M$ and $Q(x,i,j)$ is an aperiodic irreducible Markov transition matrix, that continuously depends on $x$. In particular for every $x\in M$ there exists $n_x\in\mathbb N$ such that $Q^{n_x}(x,i,j)$ is positive for every $i,j\in E$. We also define $\Phi^i$ the flow associated with the vector field $F^i$.
Next I am given $\tau>0$ and a sequence of random variables $0=T_0<T_1<\dots<T_n<...$, such that the differences $U_i:=T_i-T_{i-1}$, $i\geq 1$ are i.i.d. random variables distributed as exponentials random variables of intensity $\lambda$ and shifted by $\tau$, that is $f_{U_i}(t)=\lambda e^{-\lambda(t-\tau)}\mathbb{1}_{\{t\geq \tau\}}$. In particular, this models the fact that $U_i\geq \tau$ almost surely.
The Markov chain is then defined as follows: $Z_0=(X_0,Y_0)$ is a random variable on $M\times E$, independent of $(U_i)_{i\geq 1}$. Inductively, $X_{n+1}=\Phi^{Y_n}(U_{n+1},X_n)$ and $\mathbb P(Y_{n+1}=j\mid X_{n+1}, Y_n=i)=Q(X_{n+1},i,j)$. In this way we have defined $(Z_n)_{n\geq 0}$. The continuous version $(Z_t)$ is obtained by interpolation: namely $Z_t=X_n$ if $t\in [T_n,T_n+\tau)$, while $Z_t=(\Phi^{Y_n}(t-T_n,X_n),Y_n)$ if $t\in [T_n+\tau,T_{n+1})$. Basically, I want to force my flow to remain stopped at its initial point up to time $\tau$, and only then I follow its trajectory.
My questions:
1) Has this process been already studied in the literature? It seems to be a very specific and natural renewal (càdlàg) process, and I was guessing something should already exist; probably I don't have the right keywords to look for. In this case does it has a name?
2) What can be said about invariant (stationary) measures for this process? I've tried to compute explicitly the semigroup $P_t$, which can actually be done with precise computations, but then I started to ask myself if this actually makes sense for such a process. Since $\tau$ is positive, the Feller property for $P_t$ seems trivial, and I fear that I'm probably missing something. In particular, for the infinitesimal generator we find that $\lim_{t\to 0}\frac{P_tf-f}{t}=0$ for every $f\in C^0(M)$, and this actually makes me doubt about the construction.
I thank in advance whoever feels like providing me with some references, or giving me any hint on where to focus my research.
My bests,
Guido
