I have a question about Feller processes.

In this paper "On the doubly Feller property of resolvent" the transition semigroup $(P_t)_{t \ge 0}$ of a Hunt process (on a metric space $E$) is said to have the Feller property if the following two conditions are satisfied.

  1. For every $f \in C_{\infty}(E)$ and $t >0$, we have $P_{t}f \in C_{\infty}(E)$.
  2. For every $f \in C_{\infty}(E)$ and $x \in E$, $\lim_{t \to 0}P_{t}f(x)=f(x)$.

Here, $C_{\infty}(E)$ is the family of continuous functions on $E$ vanishing at infinity.

My question

I think the transition semigroup $(P_t)_{t \ge 0}$ of a Hunt process $X$ always satisfies the condition 2. Since $f(X_t) \to f(X_0)$ as $t \to 0$ from the right continuity of the sample path of $X$. This implies \begin{align*} \lim_{t \to 0}P_{t}f(x)=\lim_{t \to 0}E_{x}[f(X_t)]=E_{x}[f(X_0)]=E_{x}[f(x)]=f(x) \end{align*} by the dominated convergence theorem and the normal property of $X$.

Why they need the condition 2?


The Feller property is sufficient (bot not necessary) for the existence of a Hunt process associated with $(P_t)$. As you observe, condition (2) in the Feller property is (in isolation) also a necessary consequence of the Hunt property.

  • 1
    $\begingroup$ Yes: If $f\in C_\infty(E)$ then $t\mapsto f(X_t(\omega))$ is bounded and continuous for each $\omega$, with limit $f(X_0(\omega))$ as $t\downarrow 0$. Property (2) follows by dominated convergence, as you noted. $\endgroup$ – John Dawkins Jul 22 '17 at 15:30
  • $\begingroup$ Thanks for your reply. So what does Henry care about? $\endgroup$ – sharpe Jul 22 '17 at 15:34
  • $\begingroup$ Henry's concern is unclear to me. $\endgroup$ – John Dawkins Jul 22 '17 at 15:36
  • $\begingroup$ Are you saying that (2) is not necessary in definition of Hunt process...but it is a consequence of Hunt process? I am confused since in that case why the paper needs (2) in def? $\endgroup$ – Henry.L Jul 23 '17 at 0:45
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    $\begingroup$ (2) is not in the definition of Hunt process (a strong Markov process with right-continuous and quasi-left-continuous sample paths). On the other hand, (2) is a necessary consequence for the semigroup of a Hunt process. $\endgroup$ – John Dawkins Jul 23 '17 at 14:56

Sorry I probably missed "$(P_t)_{t \ge 0}$ of a Hunt process" when I first composed this answer and that is why confusion arise.

I think it is most beneficial if we start from and stick to definition.

Hunt process is a strong Markov process with some regularity on sample functions, usually càdlàg sample functions but subject to some alternations. Suppose $X$ is on a (locally) compact space $E$ and there is a one-to-one correspondence between Hunt process and Feller-Dynkin semigroups.

The paper you cited provided (1)(2) as a set of equivalent conditions that allows you to regard the transition semigroup $(P_t)_{t \ge 0}$ as a Feller-Dynkin semigroup as well. A Hunt process may admits a transition semigroup satisfying (2) of course but it also needs (1) to make the operations within the transition semingroup closed.

I hope I made myself clear now.

  • $\begingroup$ Sorry. I couldn't understand what you are saying. Why $\lim_{t \to 0}f(X_{t}(\omega))=f(\lim_{t \to 0}X_{t}(\omega))$ does not hold in general? We have continuity of $t \mapsto X_{t}(\omega)$. Do you have an counter example? $\endgroup$ – sharpe Jul 22 '17 at 6:46
  • $\begingroup$ Perhaps I have misunderstood something. $\endgroup$ – sharpe Jul 22 '17 at 13:25
  • $\begingroup$ Assume: for each $\omega$, $t \to X_{t}(\omega)$ is continuous. Then, $\lim_{t \to 0}d(X_{t}(\omega),X_{0}(\omega))=0$, where $d$ is a metric on $E$. Do you agree with this? $\endgroup$ – sharpe Jul 22 '17 at 13:30
  • $\begingroup$ What is the definition of one-sequense? $\endgroup$ – sharpe Jul 22 '17 at 14:22
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    $\begingroup$ @sharpe I will try to refine the post later and sorr for the confusion. $\endgroup$ – Henry.L Jul 23 '17 at 10:05

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