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Let $E$ be a locally compact separable metric space, $D([0,\infty),E)$ denote the Skorohod space, $C_0(E)$ denote the space of continuous functions $E\to\mathbb R$ vanishing at infinity and $E^\ast:=E\uplus\left\{\infty\right\}$ denote the Alexandroff one-point compactification.

I'm trying to understand the proof of Theorem 2.5 of Chapter 4 in the book of Ethier and Kurtz: Let

  • $(T(t))_{t\ge0}$ be a Feller (i.e. contractive, nonnegative and strongly continuous) semigroup on $C_0(E)$ with generator $(\mathcal D(A),A)$
  • $(T_n(t))_{t\ge0}$ be a uniformly continuous semigroup on the space $B(E)$ of bounded measurable functions $E\to\mathbb R$ (equipped with the supremum norm) with (bounded) generator $A_n$
  • $D$ be a core of $(\mathcal D(A),A)$
  • $(X_t)_{t\ge0}$ and $(X^n_t)_{t\ge0}$ be $E$-valued càdlàg strong Markov processes on a probability space $(\Omega,\mathcal A,\operatorname P)$ with $$\operatorname E\left[f(X_{s+t})\mid\mathcal(X_r)_{r\in[0,\:s]}\right]=(T(t)f)(X_s)\;\;\;\text{almost surely for all }f\in C_0(E)\tag1$$ and $$\operatorname E\left[f(X^n_{s+t})\mid\mathcal(X^n_r)_{r\in[0,\:s]}\right]=(T_n(t)f)(X^n_s)\;\;\;\text{almost surely for all }f\in B(E)\tag2$$ for all $s,t\ge0$ and $n\in\mathbb N$

Assume $$\left\|T_n(t)f-T(t)f\right\|_\infty\xrightarrow{n\to\infty}0\;\;\;\text{for all }f\in C_0(E)\text{ and }t\ge0\tag3,$$ which is known to be equivalent to the claim that for all $f\in D$, there is a $f_n\in B(E)$ for all $n\in\mathbb N$ with $$\left\|f_n-f\right\|_\infty+\left\|A_nf_n-Af\right\|_\infty\xrightarrow{n\to\infty}0.\tag4$$ Moreover, assume $$X_0^n\to X_0\tag5$$ in distribution.

I want to show that $(X^n_\ast\operatorname P)_{n\in\mathbb N}\subseteq\mathcal M_1(D([0,\infty),E))$ is relatively compact, where $\mathcal M_1(D([0,\infty),E))$ denotes the space of probability measures on $\mathcal B(D([0,\infty),E))$ equipped with the topology of weak convergence$^1$.

The proof in Ethier and Kurtz utilizes quite general results. I would like to proof the claim directly without using these general results. The guideline seems to be to firstly show that $(X^n_\ast\operatorname P)_{n\in\mathbb N}$ is relatively compact when considered as a sequence in $\mathcal M_1(D([0,\infty),E^\ast))$.

How can we do that?

(compare with If $X^n$ is a càdlàg process in a locally compact space $E$, show that $\{X^n\}$ is relatively compact as processes in the compactification of $E$).

EDIT: For example, in the used Theorem 9.4 of Chapter 3, we should be able to choose $C_a=C_0(E)$ and $D=\mathcal D(A)$, noting that $$f(X^n_t)-\int_0^t(A_nf)(X^n_s)\:{\rm d}s\tag6$$ is a martingale wrt the filtration $\mathcal F^{X_n}$ generated by $X^n$, and hence the conditions $(9.17)$ and $(9.18)$ should be trivially satisfied. This should allow a much simpler proof of that Theorem.

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