Suppose we are given a continuous-time (strong) Markov process $(X_t)_{t\geq 0}$ on $\mathbb R^d$, perhaps Feller if you will. For any compact subset $K\subset \mathbb R^d$ we may construct a "compactified version" $X^K$ of $X$ on $K$ in the following way: We restrict the initial values / initial distributions of $X^K$ to values in $K$. Starting with an initial value of $x_0\in K$ we set $$X^K_t:=X_{t\land \tau_K}$$ where $X$ is the original Markov process starting in $x_0$ and $\tau_K=\inf\{t\geq 0: X_t\not\in K\}$.
My question is the following:
Suppose we are given a sequence of (strong) Markov / Feller processes $X^{(n)}$, such that for all compact sets $K$ the compactified versions $(X^{(n)})^K$ converge weakly to $X^K$ in the Skorokhod-$J_1$-topology,
- Can we then say anything about the convergence of $X^{(n)}$ to $X$?
- Are there any extra assumptions that together with the given convergence of compactified processes implies weak convergence $X^{(n)}\Rightarrow X$ in the Skorokhod topology?
A similar question has been asked here in a specific case of Markov chains.
A few thoughts:
I assume weak convergence of all compactified versions is not enough to conclude weak convergence of the whole Markov process for the following reason: Typically, when working with Feller processes $X^{(n)},X$ and their semigroups $T^{(n)}_t$ and $T_t$, weak convergence of $X^{(n)}$ to $X$ is tied to strong convergence, i.e. uniform convergence of $T^{(n)}_tf$ to $T_tf$ for every $f$ in a certain function space. If a result as described above were to be true, this would (i think) entail that one could weaken the condition of uniform convergence $T^{(n)}_tf\to T_tf$ to locally uniform convergence of $T^{(n)}_tf$ to $T_tf$, i.e. uniform convergence on every compact set. However, I doubt such a simplification can be made.