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Update. Most of my questions have been answered in the comments. I am adding these answers to the post.

There are at least three definitions of Feller semigroup and the corresponding processes: $C_0 \to C_0$ (Dynkin-Feller), $C_b \to C_b$ ($C_b$-Feller) and $B \to C_b$ (strong Feller). Most of the sources I found treat the first class, which led me to wondering:

Question 1. What are exactly the properties that work for $C_0$-Feller processes, but don't for $C_b$ and why? Do I understand it right that both are strong Markov, but $C_b$-Feller may not be càdlàg?

Answer to Q1: $C_b$-Feller processes are still càdlàg, but in the Stone-Čech compactification of the state space

Question 2. What are some common examples of not $C_b$-Feller processes, which nonetheless have nice properties?

Answer to Q2: SPDE solutions are such examples, see the comment below

Question 3. It seems that if $X_x ( t )$ is $C_0$-Feller, than it has to go to infinity as $x \to \infty$, because $\mathbb{E} f ( X_x ( t ) ) \to 0$ to ensure $P_t f \in C_0$. Does this mean that Ito diffusions which are bounded with respect to initial data $X_x ( 0 ) = x$ are $C_b$-Feller but not Feller-Dynkin?

Answer to Q2 and Q3: The semigroup $P^t$ of a (deterministic) process solving $\dot{x} = -x^3$ is $C_b$-Feller, but not $C_0$-Feller. Namely, $$ (P^t f) ( x ) = f \left( \frac{1}{\sqrt{2 t +x^{-2}}} \right), $$ maps $C_b \to C_b$, but not $C_0 \to C_0$. So does the semigroup of $dx = -x^3 dt + dw$.

Question 4. There is a standard result that $C_0$-Feller property is equivalent to the following two: $X_x ( t ) \xrightarrow[]{d} X_y ( x )$ as $x \to y$ and $X_x ( t) \xrightarrow[]{\mathbb{P}} x$ as $t \to 0$ (see Lemma 17.3 in Kallenberg's Foundations or Lemmata 185 and 186 in Almost none of the Theory of Stochastic Processes, see here). Is there an analogue for $C_b$-Feller processes?

Answer to Q4: This property is equivalent to $C_b = C_0$-Feller when the state space is compact and implies $C_b$-Feller if it is not. There is a mistake in Almost None of the Theory of Stochastic Processes: the result these properties do not imply $C_0$-Feller property.

With updated understanding of the matter, here are two new questions related to Q1:

Question Q1'. Is there a way to at least ensure that $X_t$ lives in $[-\infty, \infty]$ and not the Stone-Čech compactification of $\mathbb{R}$?

Question Q1''. Where can I read about path regularity for $C_b$-Feller processes?

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  • $\begingroup$ Regarding Q2, it's not clear what sort of examples / properties you're after. Regarding Q3, what do you mean by a "bounded diffusion"? Regarding Q4, the lemma in Kallenberg you quote assumes the state space is compact, so $C_0$-Feller and $C_b$-Feller are the same (and in general that condition implies $C_b$-Feller, not $C_0$-Feller). $\endgroup$
    – Martin Hairer
    Commented Oct 24, 2023 at 16:08
  • $\begingroup$ @MartinHairer, thanks for the answer! I haven't noticed that Kallenberg changes the assumptions right before this lemma. By examples I mean something which showcases the necessity of $C_b$-Feller over $C_0$-Feller. By "bounded diffusion" I meant a solution of Ito SDE which is bounded with respect to the initial point $x_0$. $\endgroup$
    – tsnao
    Commented Oct 24, 2023 at 16:15
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    $\begingroup$ Yes, the solution to $\dot x = -x^3$ on $\mathbb{R}$ satisfies 13.3 but not 13.11, so is a counterexample to Lemma 185. I guess this may be an example of what you mean by a "bounded diffusion", if so you're right regarding Q3. I still don't understand what you mean by "necessity of $C_b$ over $C_0$ Feller", since $C_0$-Feller is the stronger property. $\endgroup$
    – Martin Hairer
    Commented Oct 24, 2023 at 16:34
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    $\begingroup$ Regarding càdlàg, this requires your space to be locally compact, or it may fail. If you're $C_b$-Feller (and in a locally compact space), you'll still have càdlàg paths, but with values in the Stone–Čech compactification of your space. Concretely, it means that the process might run off to infinity in finite time and get reinjected back into the system, which would be prevented by the $C_0$-Feller property. $\endgroup$
    – Martin Hairer
    Commented Oct 24, 2023 at 17:23
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    $\begingroup$ @tsnao For Question Q1'' in "Operator semigroups in the mixed topology and the infinitesimal description of Markov processes" econstor.eu/bitstream/10419/273041/1/1800352557.pdf, they mention the various motivations in the SPDE community for Cb and also some interesting open problems. $\endgroup$
    – Thomas Kojar
    Commented Oct 24, 2023 at 21:29

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