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I’m working with generators of Feller processes. If $C(E)$ is the space of continuous functions over $E$; with $E$ a compact metric space, I proved that an operator $G$ over $C(E)$ is the infinitesimal generator of a Feller Semigroup via Hille-Yosida theorem (I used the version in Kallenberg’s book “Foundations of Modern Probability” in chapter 17 of 2021 edition).

The fact is that we started with the process and we proposed the operator $G$. The process $X_t$ takes values in the set $E=\{1/n\colon n\in \mathbb{N}\}\cup\{0\}$, which is compact. The process starts in zero and "instantaneously" leaves that state ($\mathbb{P}_0(T=0)=1$, where $T=\inf\{t\ge0\colon X_t\not= 0\}$). In other words, has an instantaneous state, $x_0=0$. I’m asking for references that give conditions over $Gf(x_0)$ implying that $x_0$ is an instantaneous state. In other words, Can Infinitesimal Generators detect the instantaneous states in a Feller process?

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  • $\begingroup$ Could you include the definition of an instantaneous state in your question? I could only find it for discrete state spaces. $\endgroup$
    – Jochen Glueck
    Commented Jul 11, 2022 at 21:28
  • $\begingroup$ @JochenGlueck My state space is discrete. I', working with a process taking the values 1/n, for n a natural number, and zero. This implies that my state space is compact as a subset of real numbers. So, the definition for discrete spaces applies. $\endgroup$
    – Uriel Herrera
    Commented Jul 11, 2022 at 21:52
  • $\begingroup$ Interesting question (+1). Are the indicator functions $1_{\{x\}}$ for $x\not=0$ in the domain of $G$? $\endgroup$
    – Jochen Glueck
    Commented Jul 11, 2022 at 22:19
  • $\begingroup$ @JochenGlueck yes, the indicator functions $1_{\{x\}}$ are in the domain of $G$, for $x\not=0$. I was trying to use the resolvents and, in some way, make the indicator $1_{\{0\}}$ appear, but maybe this idea is not correct. $\endgroup$
    – Uriel Herrera
    Commented Jul 11, 2022 at 22:39

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