I am sure this is written down somewhere but cannot find it. Consider a Polish space $E$ and a strong Markov process $(X_t)_{t\ge 0}$ with values in $E$ and cadlag paths. More precisely, we have a family $X^x=(X^x_t)_{t\ge 0}$, $x\in E$, of Markov processes with a common transition kernel and $X^x_0=x$ almost surely. Further assume the Feller property, i.e. the corresponding semi-group maps bounded continuous functions to bounded continuous functions. Equivalently, the law of $X_t^x$ depends continuously on $x$ (for fixed $t$), hence the same is true for finite-dimensional distributions.

I know that in the case of locally compact $E$ this is enough to deduce that the law of $X^x$ depends continuously on $x$ w.r.t. weak convergence on the path-space with Skorohod topology. (see, e.g. [Ethier, Kurtz: Markov processes], Chap. 4, Thm. 2.11)

Now my question is if the assumption of local compactness is really necessary:

Does the law (in weak topology on Skorohod space) of a Feller Process on a general Polish space depend continuously on the initial condition?

If not, are there reasonable conditions to ensure this continuity in paths-pace?

I would be very thankful for any good reference for Markov processes with the Feller property on non-locally-compact spaces!

Most sources I found already include local compactness in the definition of "Feller".

I am sure this is written down somewhere but cannot find it. Consider a Polish space $E$ and a strong Markov process $(X_t)_{t\ge 0}$ with values in $E$ and cadlag paths. More precisely, we have a family $X^x=(X^x_t)_{t\ge 0}$, $x\in E$, of Markov processes with a common transition kernel and $X^x_0=x$ almost surely. Further assume the Feller property, i.e. the corresponding semi-group maps bounded continuous functions to bounded continuous functions. Equivalently, the law of $X_t^x$ depends continuously on $x$ (for fixed $t$), hence the same is true for finite-dimensional distributions.

I know that in the case of locally compact $E$ this is enough to deduce that the law of $X^x$ depends continuously on $x$ w.r.t. weak convergence on the path-space with Skorohod topology (Edit: it is not, see below).

Now my question is if the assumption of local compactness is really necessary:

Does the law (in weak topology on Skorohod space) of a Feller Process on a general Polish space depend continuously on the initial condition?

If not, are there reasonable conditions to ensure this continuity in paths-pace?

Edit: As George pointed out, contrary to my claim the above does not even hold for locally compact spaces if we do not assume that the functions vanish at $\infty$, which makes no sense in non-locally-compact spaces. Still it would be nice to have a reference about processes with the continuity property defined above.

I would be very thankful for any good reference for Markov processes with the Feller property on non-locally-compact spaces!

Most sources I found already include local compactness in the definition of "Feller".