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Let $E$ be a locally compact and separable metric space, and suppose $X$ is a Feller process with transition function $P_t$. To be precise, let $C_0$ denote the space of continuous functions vanishing at infinity, suppose $P_tf\in C_0$ whenever $f\in C_0$, and $$\sup_{x\in E}|P_tf(x)- f(x)|\rightarrow 0$$ as $t\rightarrow 0$.

Let $P_x$ denote the law of the process started at $x\in E$. It is stated on page 625 of this paper that the laws $\{P_x\}_{x\in E}$ are continuous in the space of cadlag paths with the Skorohod topology, by this I mean that if $x_n\rightarrow x$ in $E$, then $P_{x_n}$ converges weakly to $P_x$ as measures on Skorohod space. A more general question is very briefly discussed here, however I've been to unable to find a reference.

Do you know of a book or paper where this is proved?

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  • $\begingroup$ While I find this question interesting (and I do not know the answer off the top of my head), if I am not mistaken, the authors only claim weak continuity of laws. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Apr 13, 2019 at 9:07
  • $\begingroup$ Yes you are right, I really mean continuous in the same sense as the authors, that is, if $x_n\rightarrow x$ in $E$, then $P_{x_n}\rightarrow P_x$ weakly as measures in Skorohod space. I've edited the question to make this clearer. Thanks for your input. $\endgroup$
    – Potato
    Commented Apr 13, 2019 at 17:25
  • $\begingroup$ I just stumbled upon a paper where, as far as I understand, a reference for the result you ask is Theorem 4.2.5 in: Ethier, Kurtz, Markov processes: characterization and convergence, Wiley, 1986. I did not check this, though. $\endgroup$
    – Mateusz Kwaśnicki
    Commented May 15, 2019 at 13:43

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