Revision 41b43af8-bc6d-4cfc-acfd-0b81aeba7633

Yes, it is possible to extend the state space with respect to which $Y$ is a Feller process. Then, $X$ will be a dense open subset of the extension $\hat X$. Furthermore, for any initial distribution of $Y_0\in\hat X$, then $Y$ will have a continuous modification which necessarily satisfies $Y_t\not\in\hat X\setminus X$ for all positive times (almost surely). In your example with $Y$ being a Brownian motion and $\hat X=\mathbb{R}^3\setminus\{0\}$ then this process corresponds to adding back the origin.

This is actually a special case of a more general method called *Ray-Knight compactification*, which applies to right-continuous Markov processes taking values in a [Suslin space][1]. Ray-Knight compactification does not always lead to processes which are Feller though, as they can have branch points in the extended state space. The type of processes obtained by this method are called *Ray processes*. See [the 1975 paper][2] by Getoor & Sharpe detailing Ray-Knight compactification, or, any reasonably comprehensive textbook on Markov processes should describe this method. In your question the conditions that $X$ is locally compact, $T_tC_0\subseteq C_b$ and $Y$ is continuous are enough to ensure that we obtain a Feller process so, in particular, there will be no branch points in the extended domain. One point before continuing; I'm assuming, as is standard in the definition of Feller processes, that the space $X$ has a countable base (locally compact, Hausdorff with a countable base, aka an *lccb* space).

Constructing the extension is a rather natural application of the [Gelfand-Naimark theorem][3], although showing that the resulting process is Feller will still require some work. Let $\mathcal{A}$ be the smallest closed $C^*$-subalgebra of $C_b(X)$ containing $C_0(X)$ and closed under application of $T_t$. We can define $\hat X$ to be its spectrum (the set of nonzero $C^*$ homomorphisms $\mathcal{A}\to\mathbb{C}$), under the weak topology. Then, we identify $X$ as an (open, dense) subset of the (locally compact) space $\hat X$ in the same way as in the construction of the [Stone-Chech compactification][4]. By the Gelfand-Naimark theorem, every $f\in\mathcal{A}$ extends uniquely to an $\hat f\in C_0(\hat X)$, giving an isometry $\mathcal{A}\to C_0(\hat X)$, $f\mapsto\hat f$. Then, $T_t$ extends to a map $\hat T_t\colon C_0(\hat X)\to C_0(\hat X)$, $\hat T_t\hat f=\hat{T_tf}$. This is automatically a Markov transition function, but we can show that it is also Feller in your case. It is a bit of work, and I'll break it up into smaller statements.

> $\hat X$ is an lccb space.

As $X$ is lccb, $C_0(X)$ has a countable dense subset $S$. Then, the set of rational linear combinations of products of terms $\{T_t f\colon t\in\mathbb{Q}\_+,f\in S\}$ is a countable dense subset of $\mathcal{A}$, and $C_0(\hat X)\cong\mathcal{A}$ is separable. This implies that $\hat X$ has a countable base.

> $T_tC_b(X)\subseteq C_b(X)$.

Any nonnegative $f\in C_b(X)$ is a pointwise limit of an increasing sequence $f_n\in C_0(X)$ and, therefore $T_tf=\lim\_{n\to\infty}T_tf_n$ is also a limit of an increasing sequence in $C_0(X)$, so is [lower semicontinuous][5]. Applying the same statement to $\Vert f\Vert - f$ implies that $f$ is also upper semi-continuous, so is continuous.

> For any $f\in C_b(X)$ and $t_n\to0$, $T_{t_n}f\to f$ in the compact-open topology.

This is a fairly standard kind of argument showing that weak continuity of semigroups implies strong continuity. By right-continuity of the process $Y$, we have $T_{t_n}f(x)\to f(x)$ as $n\to\infty$ for all $x\in X$. Then, setting $g_m=m\int\_0^{1/m}T_sf\\,ds$, it can be seen that $\Vert T_{t_n}g_m-g_m\Vert\le 2mt_n\Vert f\Vert\to0$ as $n\to\infty$. Now, letting $S_m$ be the space of convex combinations of $\{g_m,g_{m+1},\ldots\}$, $f$ is a pointwise limit of a sequence in $S_m$, so is in its closure under the compact-open topology (this is a consequence of the Hahn-Banach theorem and the [Riesz representation theorem][6]). Therefore, there exists $f_m\in S_m$ converging to $f$ in the compact-open topology. However, by construction, $T_{t_n}f_m\to f_m$ uniformly as $n\to\infty$ and, taking the limit $m\to\infty$, we can conclude that $T_{t_n}f\to f$ under the compact-open topology (you do need to show that you can commute the limits like this, but it's not too hard an exercise).

> For any $f\in\mathcal{A}$ and $t_n\to0$, $T_{t_n}f\to f$ *uniformly* as $n\to\infty$.


This is the point where continuity is required. It is not too hard to show that the space of $f\in C_b(X)$ satisfying the conclusion forms a $C^*$-algebra and is closed under $T_t$, so it is enough to prove it for $f\in C_b(X)$ with compact support $S$. Also, the fact that $T_tC_0(X)\subseteq C_b(X)$ can be used to prove the strong-Markov property. So, letting $\tau$ be the first time at which $Y$ hits $S$, continuity implies that $f(Y_{\tau})=f(Y_0)$ and,
$$
\begin{align}
T_tf(x)-f(x)=\mathbb{E}\_x[f(Y_t)-f(Y_0)]&=\mathbb{E}\_x\left[T_{(t-\tau)\_+}f(Y_{\tau\wedge t})-f(Y_{\tau\wedge t})\right]\\\\
&\le\sup\_{y\in S,s\le t}\left\vert T_sf(y)-f(y)\right\vert.
\end{align}
$$
The previous statement says that the right-hand-side tends to zero as $t\to0$ and, as it is independent of $x$, it bounds $\Vert T_tf-f\Vert$.

So, $\hat T_t$ is a Feller transition function on $\hat X$. Therefore, any Markov process $\hat Y$ w.r.t. this transition function has a right-continuous modification. We can also show that, regardless of the initial distribution, $\hat Y$ will be continuous and $\hat Y_t\not\in\hat X\setminus X$ for all times $t > 0$ (almost-surely). For any time $s > 0$ at which $Y_s\in X$ then the statement of the question implies that $\hat Y$ has a continuous modification lying in $X$ at all times $t\ge s$ (which is unique). So, it is enough to show that $\mathbb{P}(\hat Y_s\in X)=1$ for each time $s > 0$. But, this is standard (see Prop. 4.6 (iii) from the [paper][2] of Getoor & Sharpe).

  [1]: http://en.wikipedia.org/wiki/Souslin_space#Suslin_spaces
  [2]: http://www.numdam.org/item?id=AIF_1975__25_3-4_207_0
  [3]: http://en.wikipedia.org/wiki/Gelfand_representation#Statement_of_the_commutative_Gelfand-Naimark_theorem
  [4]: http://en.wikipedia.org/wiki/Stone-Chech_compactification#Construction_using_C.2A-algebras
  [5]: http://en.wikipedia.org/wiki/Semi-continuity
  [6]: http://en.wikipedia.org/wiki/Riesz_representation_theorem#The_representation_theorem_for_the_dual_of_C0.28X.29