On the Restriction of a Right Process Outside a Negligible Set

Abstract

The objective of this paper is to examine the restriction of a right process on a Radon topological space, excluding a negligible set, and investigate whether the restricted object can induce a Markov process with desirable properties. We address this question in three aspects: the induced process necessitates only right continuity; it is a right process, and the semi-Dirichlet form of the induced process is quasi-regular. The main findings characterize the negligible set that meets the requirements within a universally measurable framework. These characterizations can be employed to generate instances of Markov processes that are non-right or (semi-)Dirichlet forms that are non-quasi-regular. Specifically, we will construct an example of a non-tight, strong Feller, symmetric right process on a non-Lusin Radon topological space, whose Dirichlet form is not quasi-regular.

Appendices

Appendix A: Basics of Right Processes

1.1 A.1 Radon Space

Let E be a Radon topological space, i.e. it is homeomorphic to a universally measurable subset of a compact metric space. The Borel $\sigma$-algebra on E is denoted by $\mathcal {B}(E)$, and the universal completion of $\mathcal {B}(E)$ is denoted by $\mathcal {B}^u(E)$, i.e.

$$\begin{aligned} {\mathcal {B}}^u(E)=\cap _{\mu } \overline{{\mathcal {B}}(E)}^\mu , \end{aligned}$$

where $\overline{{\mathcal {B}}(E)}^\mu$ is the completion of ${\mathcal {B}}(E)$ with respect to $\mu$ running over all finite positive measures on $(E,\mathcal {B}(E))$. Note that every finite measure on $(E,\mathcal {B}(E))$ extends in a unique way to a measure on $(E,\mathcal {B}^u(E))$, and every finite measure on $(E,\mathcal {B}^u(E))$ is the unique extension of its restriction to ${\mathcal {B}}(E)$; see, e.g., [16, (A1.1)]. Hence we would also call $\mu$ just a measure on E if no confusions are caused. The simpler notations ${\mathcal {E}}$ and ${\mathcal {E}}^u$ in place of ${\mathcal {B}}(E)$ and ${\mathcal {B}}^u(E)$ will be used unless clarity dictates otherwise.

Suppose E is embedded into the compact metric space $\hat{E}$ with the metric d compatible to the topology on E, i.e. the subspace topology of E relative to $(\hat{E}, d)$ is identical to the original topology of E. Note that ${\mathcal {E}}={\mathcal {B}}(\hat{E})|_E:=\{E\cap \hat{B}: \hat{B}\in {\mathcal {B}}(\hat{E})\}$ and ${\mathcal {E}}^u={\mathcal {B}}^u(\hat{E})|_E:=\{E\cap \hat{B}: \hat{B}\in {\mathcal {B}}^u(\hat{E})\}$; see, e.g., [16, (A2.2)]. We set $C(E)=C(E,d)$ to be the family of all real continuous functions on E and $C_d(E)$ to be the family of all real d-uniformly continuous functions on E. Then $C_d(E)$ is a separable space with respect to the uniform norm, while C(E) may be not.

1.2 A.2 Filtered Measurable Space

Given a $\sigma$-algebra $\mathcal {M}$ on a space M, $b\mathcal {M}$ (resp. $p\mathcal {M})$) stands for the class of bounded (resp. $[0,\infty ]$-valued) $\mathcal {M}$-measurable functions on M. Given two measurable spaces $(M_1,\mathcal {M}_1)$ and $(M_2,\mathcal {M}_2)$. A map $f: M_1\rightarrow M_2$ is measurable, denoted by $f\in \mathcal {M}_1/\mathcal {M}_2$, if $f^{-1}(B_2)\in \mathcal {M}_1$ for any $B_2\in \mathcal {M}_2$.

Let $(\Omega , {\mathcal {G}}, \textbf{P})$ be a probability space and $(X_t)_{t\ge 0}$ be a stochastic process with values in E. That is, $X_t$, $t\ge 0$, is a collection of measurable maps from $(\Omega , {\mathcal {G}})$ to $(E,{\mathcal {E}})$. To emphasize the dependence on ${\mathcal {E}}$, we call it an ${\mathcal {E}}$-stochastic process. Similar definitions will apply when ${\mathcal {E}}$ is replaced by larger $\sigma$-algebra ${\mathcal {E}}^\bullet$, e.g., ${\mathcal {E}}^u$. In the current paper a stochastic process tacitly means an ${\mathcal {E}}^u$-stochastic process. It is required in this case that for any $t\ge 0$, $\{X_t\in B\}:=\{\omega \in \Omega : X_t(\omega )\in B\}\in {\mathcal {G}}$ for every $B\in {\mathcal {E}}^u$ rather than every $B\in {\mathcal {E}}$. A filtration $({\mathcal {G}}_t)$ means an increasing family of sub-$\sigma$-algebras of ${\mathcal {G}}$, to which $(X_t)$ is (${\mathcal {E}}^u$-)adapted in the sense that $X_t\in {\mathcal {G}}_t/{\mathcal {E}}^u$ for $t\ge 0$. Then the collection $(\Omega , {\mathcal {G}}_t,{\mathcal {G}})$ is called a filtered measurable space.

Corresponding to a fixed ${\mathcal {E}}^u$-stochastic process $(X_t)$ on $\Omega$, the natural $\sigma$-algebra ${\mathcal {F}}^u_t$ is defined as $\sigma \left\{ f(X_s): 0\le s\le t, f\in {\mathcal {E}}^u\right\}$ and ${\mathcal {F}}^u:=\sigma \left\{ f(X_t): t\ge 0, f\in {\mathcal {E}}^u\right\}$. Obviously ${\mathcal {F}}^u_t\subset {\mathcal {G}}_t$ and ${\mathcal {F}}^u\subset {\mathcal {G}}$.

1.3 A.3 Transition Function

Fix a $\sigma$-algebra ${\mathcal {E}}^\bullet$ on E with ${\mathcal {E}}\subset {\mathcal {E}}^\bullet \subset {\mathcal {E}}^u$. The symbol K(x, dy) is a kernel on $(E,{\mathcal {E}}^\bullet )$ provided that, for all $x\in E$, K(x, dy) is a positive measure on $(E,{\mathcal {E}}^\bullet )$, and for every $B\in {\mathcal {E}}^\bullet$, $x\mapsto K(x,B)$ is ${\mathcal {E}}^\bullet$-measurable. It is called a Markov (sub-Markov) kernel if $K(x,E)=1$ (resp. $K(x,E)\le 1$) for all $x\in E$. Note that K can be always extended to a kernel on $(E,{\mathcal {E}}^u)$, which is still denoted by K.

Definition A.1

A family $(P_t)_{t\ge 0}$ of Markov kernels on $(E,{\mathcal {E}}^\bullet )$ is called a transition function on $(E,{\mathcal {E}}^\bullet )$ if for all $t,s\ge 0$ and all $x\in E, B\in {\mathcal {E}}^\bullet$,

$$\begin{aligned} P_{t+s}(x,B)=\int _E P_t(x,dy)P_s(y,B). \end{aligned}$$

(A.1)

It is called a normal transition function if in addition, $P_0(x, B)=1_B(x)$ for all $x\in E$ and all $B\in {\mathcal {E}}^\bullet$.

Let $(X_t)$ be a stochastic process on $(\Omega , {\mathcal {G}},\textbf{P})$ ${\mathcal {E}}^\bullet$-adapted to $({\mathcal {G}}_t)$. It satisfies the Markov property relative to a transition semigroup $(P_t)$ on $(E,{\mathcal {E}}^\bullet )$ provided that

$$\begin{aligned} \textbf{P}\{f(X_{t+s})|{\mathcal {G}}_t\}=P_sf(X_t),\quad t,s\ge 0, f\in b{\mathcal {E}}^\bullet . \end{aligned}$$

(A.2)

The transition function $(P_t)$ is also called the (Markov) transition semigroup of $(X_t)$ due to the semigroup property Eq. A.1. The distribution $\mu$ of $X_0$ is called the initial law of $(X_t)$. Clearly $\mu _t=\mu P_t$ is the distribution of $X_t$.

1.4 A.4 The First Regularity Hypothesis (HD1)

Let $(X_t)_{t\ge 0}$ be a stochastic process defined on $(\Omega , {\mathcal {G}},\textbf{P})$ and having values in a topological space E. It is called right continuous in case that every sample path $t\mapsto X_t(\omega )$ is a right continuous map from $\mathbb {R}^+:=[0,\infty )$ to E. The following hypothesis, due to [16, (2.1)], is essentially the first of Meyer’s hypothèses droites.

Definition A.2

(HD1) Let E be a Radon topological space. A Markov transition function $(P_t)_{t\ge 0}$ on $(E, {\mathcal {E}}^u)$ is said to satisfy (HD1), if given an arbitrary probability law $\mu$ on E, there exists an E-valued right continuous stochastic process $(X_t)_{t\ge 0}$ on some filtered probability space $(\Omega , {\mathcal {G}},{\mathcal {G}}_t, \textbf{P})$ so that $X=(\Omega , {\mathcal {G}},{\mathcal {G}}_t,\textbf{P}, X_t)$ satisfies the Markov property Eq. A.2 (${\mathcal {E}}^\bullet ={\mathcal {E}}^u$) with transition semigroup $(P_t)_{t\ge 0}$ and initial law $\mu$.

In order to facilitate computations we shall work with a fixed collection of random variables $X_t$ defined on some probability space, and a collection $\textbf{P}^x$ specified in such a way that $\textbf{P}^x(X_0=x)=1$, and under every $\textbf{P}^x$, $X_t$ is Markov with semigroup $(P_t)$. That is the following.

Definition A.3

Let E be a Radon space and $(P_t)_{t\ge 0}$ be a Markov transition function satisfying (HD1). The collection $X=(\Omega , {\mathcal {G}},{\mathcal {G}}_t, X_t,\theta _t, \textbf{P}^x)$ is called a realization of $(P_t)$ if it satisfies the following conditions:

• (1) $(\Omega , {\mathcal {G}},{\mathcal {G}}_t)$ is a filtered measurable space, and $X_t$ is an E-valued right continuous process ${\mathcal {E}}^u$-adapted to $({\mathcal {G}}_t)$;

• (2) $(\theta _t)_{t\ge 0}$ is a collection of shift operators mapping $\Omega$ into itself and satisfying for $t,s\ge 0$, $\theta _t\circ \theta _s=\theta _{t+s}$ and $X_t\circ \theta _s=X_{t+s}$;

• (3) For every $x\in E$, $\textbf{P}^x(X_0=x)=1$, and the process $(X_t)_{t\ge 0}$ has the Markov property Eq. A.2 with transition semigroup $(P_t)$ relative to $(\Omega ,{\mathcal {G}},{\mathcal {G}}_t, \textbf{P}^x)$.

Furthermore, a realization of $(P_t)$ is called canonical if $\Omega$ is the space of all right continuous maps from $\mathbb {R}^+$ to E, $X_t(\omega ):=\omega (t)$, ${\mathcal {G}}=\sigma \left\{ f(X_t): f\in {\mathcal {E}}^u,t\ge 0\right\}$ and ${\mathcal {G}}_t=\sigma \left\{ f(X_s): f\in {\mathcal {E}}^u,0\le s\le t\right\}$.

Remark A.4

• (1) The normal property $\textbf{P}^x(X_0=x)=1$ is not always built into the definition of Markov property Eq. A.2. Particularly, it implies that the transition semigroup $(P_t)$ is normal.

• (2) Obviously $\theta _t\in {\mathcal {F}}^u_{t+s}/{\mathcal {F}}^u_s$ for any $s\ge 0$.

• (3) The existence of a realization is equivalent to (HD1) for a normal transition function $(P_t)$. Such a (canonical) realization of a normal $(P_t)$ satisfying (HD1) always exists. More precisely, define $\theta _t\omega (s):=\omega (t+s)$ for $t,s\ge 0$ and $\omega \in \Omega$, the space of all right continuous maps from $\mathbb {R}^+$ to E. For $x\in E$, let $(\tilde{\Omega }, {\mathcal {G}},{\mathcal {G}}_t, \textbf{P}, \tilde{X}_t)$ be the collection in Definition A.2 with $\mu =\delta _x$. Define a map $\Phi : \tilde{\Omega }\rightarrow \Omega$, $\tilde{\omega }\mapsto \Phi (\tilde{\omega })$ with $\Phi (\tilde{\omega })(t):=\tilde{X}_t(\tilde{\omega })$, as is characterized by the formulae $\tilde{X}_t=X_t\circ \Phi$, $t\ge 0$. Obviously $\Phi \in {\mathcal {G}}/{\mathcal {F}}^u$. Let $\textbf{P}^x$ be the image measure of $\textbf{P}$ under the map $\Phi$, so that $X=(\Omega , {\mathcal {F}}^u,{\mathcal {F}}^u_t, X_t,\theta _t, \textbf{P}^x)$ is the canonical realization of $(P_t)$. Conversely, let $X=(\Omega , {\mathcal {G}},{\mathcal {G}}_t, X_t,\theta _t, \textbf{P}^x)$ be a realization of $(P_t)_{t\ge 0}$. Note that $x\mapsto \textbf{P}^x (B)$ is ${\mathcal {E}}^u$-measurable for every $B\in {\mathcal {F}}^u$; see [16, (2.6)]. Hence $\textbf{P}^\mu (\cdot ):=\int _E \mu (dx)\textbf{P}^x(\cdot )$ defines a probability measure on $(\Omega ,{\mathcal {F}}^u)$ for any probability measure $\mu$ on E, and $(X_t)$ satisfies the Markov property relative to $(\Omega , {\mathcal {F}}^u,{\mathcal {F}}^u_t,\textbf{P}^\mu )$, with transition semigroup $(P_t)$ and initial law $\mu$. In other words, $(P_t)$ satisfies (HD1).

1.5 A.5 Augmented Filtration

Now we introduce the notation of augmentation of the natural filtration ${\mathcal {F}}^u_t$ (not necessarily on the space of right continuous maps). The augmentation of general filtration is analogous.

Given an initial law $\mu$ on E, let ${\mathcal {F}}^\mu$ denote the completion ${\mathcal {F}}^u$ relative to $\textbf{P}^\mu$, and let $\mathcal {N}^\mu$ denote the $\sigma$-ideal of $\textbf{P}^\mu$-null sets in ${\mathcal {F}}^\mu$. Define ${\mathcal {F}}:=\cap _\mu {\mathcal {F}}^\mu$, where $\mu$ runs over all initial laws on E, $\mathcal {N}:=\cap _\mu \mathcal {N}^\mu$, ${\mathcal {F}}^\mu _t:={\mathcal {F}}^u_t \vee \mathcal {N}^\mu$ and ${\mathcal {F}}_t:=\cap _\mu {\mathcal {F}}^\mu _t$. Two random variables $G,H\in {\mathcal {F}}$ are called a.s. equal if $\{G\ne H\}\in \mathcal {N}$. The filtration $({\mathcal {F}}_t)$ is called the augmented natural filtration on $\Omega$.

Let $X=(\Omega ,{\mathcal {G}},{\mathcal {G}}_t,X_t,\theta _t, \textbf{P}^x)$ be a realization of $(P_t)$ as in Definition A.3. Then $\theta _t\in {\mathcal {F}}_{t+s}/{\mathcal {F}}_{s}$ for any $s\ge 0$, and $(\Omega , {\mathcal {F}},{\mathcal {F}}_t,X_t,\theta _t,\textbf{P}^x)$ is also a realization of $(P_t)$.

1.6 A.6 The Second Fundamental Hypothesis

We assume throughout this subsection that $X=(\Omega ,{\mathcal {G}},{\mathcal {G}}_t, X_t, \theta _t, \textbf{P}^x)$ is a right continuous Markov process with transition semigroup $(P_t)$ on a Radon space E.

The resolvent $(U^\alpha )_{\alpha \ge 0}$ is the family of kernels on $(E,{\mathcal {E}}^u)$ defined by

$$\begin{aligned} U^\alpha f(x):=\textbf{P}^x \int _0^\infty e^{-\alpha t}f(X_t)dt,\quad \alpha \ge 0, f\in p {\mathcal {E}}^u. \end{aligned}$$

It satisfies the well-known resolvent equation: For $0\le \alpha \le \beta$ and $f\in p{\mathcal {E}}^u$,

$$\begin{aligned} U^\alpha f=U^\beta f+(\beta -\alpha ) U^\alpha U^\beta f. \end{aligned}$$

The family of $\alpha$-excessive functions is crucial in the theory of Markov processes.

Definition A.5

Let $\alpha \ge 0$ and $f\in p{\mathcal {E}}^u$. Then f is called $\alpha -super-mean-valued$ in case $e^{-\alpha t}P_t f\le f$ for all $t\ge 0$, and $\alpha -excessive$ if in addition, $e^{-\alpha t}P_t f\rightarrow f$ as $t\downarrow 0$. It is called simply excessive if f is 0-excessive. The classes of $\alpha$-excessive, excessive functions are denoted by ${\mathcal {S}}^\alpha$, ${\mathcal {S}}$ respectively.

Remark A.6

A function $f\in p{\mathcal {E}}^u$ is called $\alpha -supermedian$ in case $\beta U^{\alpha +\beta } f\le f$ for all $\beta >0$. Note that $\alpha$-super-mean-valued functions are $\alpha$-supermedian, but not vice versa. In addition, ${\mathcal {S}}^\alpha =\{f \text { is }\alpha \text {-supermedian}: \lim _{\beta \uparrow \infty }\beta U^{\alpha +\beta }f=f\}$.

We take up now the second fundamental hypothesis, which, in particular, implies the strong Markov property.

Definition A.7

(HD2) The Markov process $X=(\Omega ,{\mathcal {G}},{\mathcal {G}}_t, X_t, \theta _t, \textbf{P}^x)$ with transition semigroup $(P_t)$ is said to satisfy (HD2), if for every $\alpha >0$ and every $f\in {\mathcal {S}}^\alpha$, the process $t\mapsto f(X_t)$ is a.s. right continuous.

Remark A.8

In view of (A.5), “a.s." means that there is $N\in \mathcal {N}({\mathcal {G}})$ such that $t\mapsto f(X_t)$ is right continuous on $\Omega \setminus N$, where $\mathcal {N}({\mathcal {G}})$ is the intersection of all $\sigma$-ideals of $\textbf{P}^\mu$-null sets in the completion of ${\mathcal {G}}$ relative to $\textbf{P}^\mu$. Briefly speaking, it says that for any $x\in E$, $t\mapsto f(X_t)$ is $\textbf{P}^x$-a.s. right continuous. This hypothesis always implies the strong Markov property of X in the sense of [16, §6]. Particularly, if E is Lusin, i.e. it is homeomorphic to a Borel subset of a compact metric space, and $P_t(b{\mathcal {E}})\subset b{\mathcal {E}}$, then (HD2) is equivalent to the strong Markov property of X.

1.7 A.7 Right Processes and Right Semigroups

Now we are in a position to formulate the definition of right processes.

Definition A.9

A system $X=(\Omega , {\mathcal {G}},{\mathcal {G}}_t,X_t,\theta _t,\textbf{P}^x)$ is a right process on the Radon space E with transition semigroup $(P_t)$ provided:

• (i) X is a realization of $(P_t)$;

• (ii) X satisfies (HD2);

• (iii) $({\mathcal {G}}_t)$ is augmented and right continuous.

If there is some right process with transition semigroup $(P_t)$, then $(P_t)$ is called a right semigroup.

Remark A.10

• (1) When E is a Lusin topological space and $P_t(b{\mathcal {E}})\subset b{\mathcal {E}}$, X is called a Borel right process.

• (2) If X is a right process, then the augmented natural filtration ${\mathcal {F}}_t$ is right continuous and $(\Omega , {\mathcal {F}},{\mathcal {F}}_t, X_t, \theta _t,\textbf{P}^x)$ is also a right process with transition semigroup $(P_t)$. Particularly, if $(P_t)$ is a right semigroup, then its augmented canonical realization, obtained by replacing the natural filtration in the canonical realization with the augmented one, is a right process with transition semigroup $(P_t)$.

Let X be a right process on E with right semigroup $(P_t)$. The fine topology (see [16, §10]) is the coarsest topology on E making all functions in $\cup _{\alpha > 0}{\mathcal {S}}^\alpha$ continuous. It is finer than the original topology of E. The Borel $\sigma$-algebra relative to the fine topology on E is denoted by ${\mathcal {E}}^e$. Actually ${\mathcal {E}}\subset {\mathcal {E}}^e=\sigma \{\cup _{\alpha >0} {\mathcal {S}}^\alpha \}$ and $P_t(b{\mathcal {E}}^e)\subset b{\mathcal {E}}^e$. For any $B\in {\mathcal {E}}^e$ (more generally, if B is nearly optional in the sense of [16, (5.1)]), the first hitting time $T_B:=\inf \{t>0: X_t\in B\}$ is an ${\mathcal {F}}_{t}$-stopping time, i.e. $\{T_B\le t\}\in {\mathcal {F}}_t$ for any $t\ge 0$.

1.8 A.8 Lifetime

If a transition function $(P_t)$ is only sub-Markovian, it may be extended to a Markov one $(\tilde{P}_t)$ on a larger space $E_\Delta$ by a standard argument as in [16, (11.1)]. (Take an abstract point $\Delta$ not in E and let $E_\Delta :=E\cup \{\Delta \}$ be the Radon space obtained by adjoining $\Delta$ to E as an isolated point.) In this case we call $(P_t)$ a right semigroup if $(\tilde{P}_t)$ is a right semigroup.

Realizing the right semigroup $(\tilde{P}_t)$ as a right process $(\Omega , \tilde{{\mathcal {G}}},\tilde{{\mathcal {G}}}_t,\tilde{X}_t, \tilde{\theta }_t,\tilde{\textbf{P}}^x)$ on $E_\Delta$, one gets $\tilde{\textbf{P}}^\Delta (\tilde{X}_t=\Delta , \forall t\ge 0)=1$. Let $\zeta :=\{t>0:\tilde{X}_t=\Delta \}$. By the strong Markov property, $\tilde{X}_t=\Delta$ for all $t>\zeta$, almost surely. Hence $\Delta$ is usually called the cemetery for the process and $\zeta$ is called the lifetime. The role played by $\tilde{P}_t$ is de-emphasized by making the convention that every function on E is automatically extended to $E_\Delta$ by setting $f(\Delta ):=0$. Defining $X_t$ to be the same as $\tilde{X}_t$ on $\Omega$ and letting $\textbf{P}^x:=\tilde{\textbf{P}}^x$ for $x\in E$, we can obtain another collection $(\Omega , {\mathcal {G}},{\mathcal {G}}_t,X_t,\theta _t, \textbf{P}^x)$ on E in a certain standard manner, which is called the right process on E with lifetime $\zeta$ and transition semigroup $(P_t)$. More details can be found in [16, §11].

Appendix B: Ray-Knight Compactification

Let $X=(\Omega , {\mathcal {G}},{\mathcal {G}}_t,X_t,\theta _t,\textbf{P}^x)$ be a right process on a Radon space E with right semigroup $(P_t)$. The resolvent of X is denoted by $(U^\alpha )_{\alpha \ge 0}$. If X has a cemetery, it should be regarded as a point in E throughout this section. The following introduction to Ray-Knight compactification is due to [16, §9, §17, §18, §39] and [8].

Let $\mathbb {Q}$ denote the set of rational numbers, $\mathbb {Q}^+$ the positive rational numbers and $\mathbb {Q}^{++}$ the strictly positive rational numbers. A family $\mathcal {Y}$ of functions is called a $\mathbb {Q}^+-cone$, if $c_1 f+c_2 g\in \mathcal {Y}$ for all $c_1,c_2\in \mathbb {Q}^+$ and $f,g\in \mathcal {Y}$. (In more standard terminology, a $\mathbb {Q}^+$-cone is known as a convex cone with respect to the field $\mathbb {Q}$ of rational numbers.) The $\mathbb {Q}^+$-cone generated by a family $\mathcal L$ of positive and bounded functions on E is defined as the smallest $\mathbb {Q}^+$-cone that contains $\mathcal L$. It is actually the set of all $\mathbb {Q}^+$-linear combinations of functions in the class $\mathcal L$. Given a $\mathbb {Q}^+$-cone $\mathcal {Y}\subset bp{\mathcal {E}}^u$, set

$$\begin{aligned} \begin{aligned}&\bigwedge (\mathcal {Y}):=\{k_1\wedge \cdots \wedge k_n: n\ge 1, k_1,\cdots , k_n\in \mathcal {Y}\}, \\&\mathcal {U}(\mathcal {Y}):=\{U^{\alpha _1}k_1+\cdots +U^{\alpha _n}k_n: n\ge 1,\alpha _i\in \mathbb {Q}^{++},k_i\in \mathcal {Y}\}. \end{aligned} \end{aligned}$$

Both operations of $\bigwedge$ and $\mathcal {U}$ keep the property of a $\mathbb {Q}^+$-cone.

Recall that $C_d(E)$ is the family of all d-uniformly continuous functions on E.

For convenience, we propose to assign a name to the following class of functions.

Definition B.1

A family ${\mathcal {C}}$ is called a pre-Ray class, if

• (i) ${\mathcal {C}}\subset p C_d(E)$ is countable;

• (ii) $1_E\in {\mathcal {C}}$;

• (iii) The linear span of ${\mathcal {C}}$ is uniformly dense in $C_d(E)$.

Since $C_d(E)$ is separable, such ${\mathcal {C}}$ always exists. The rational Ray cone $\mathcal {R}$ generated by $(U^\alpha )$ and ${\mathcal {C}}$ is the $\mathbb {Q}^+$-cone defined as follows: Let $\mathcal {H}$ denote the $\mathbb {Q}^+$-cone generated by ${\mathcal {C}}$, and let $\mathcal {R}_0:=\mathcal {U}(\mathcal {H})$. For $n\ge 1$, let $\mathcal {R}_n:=\bigwedge (\mathcal {R}_{n-1}+\mathcal {U}(\mathcal {R}_{n-1}))$, and finally set $\mathcal {R}:=\cup _{n\ge 0}\mathcal {R}_n$. Obviously $\mathcal {R}\subset \cup _{\alpha >0} b {\mathcal {S}}^\alpha$, and $\mathcal {R}$ is countable, stable under the operation $\bigwedge$, contains the positive rational constant functions, and separates the points of E.

Write $\mathcal {R}=\{g_n:n\ge 1\}$. Define a metric $\rho$ on E as

$$\begin{aligned} \rho (x,y):=\sum _{n\ge 1}2^{-n}\Vert g_n\Vert ^{-1}|g_n(x)-g_n(y)|, \end{aligned}$$

where $\Vert g_n\Vert :=\sup _{x\in E}|g_n(x)|$. The map

$$\begin{aligned} \Psi : E\rightarrow K:=\prod _{n=1}^\infty [0, \Vert g_n\Vert ], \quad x\mapsto (g_n(x))_{n\ge 1} \end{aligned}$$

is an injection. Since the product topology on K is generated by the metric

$$\rho '(a,b):=\sum _{n\ge 1}2^{-n}\Vert g_n\Vert ^{-1}|a_n-b_n|$$

for $a=(a_n)_{n\ge 1}$ and $b=(b_n)_{n\ge 1}$, $\Psi$ is an isometry of $(E,\rho )$ to $(K,\rho ')$. It follows that the completion $(\bar{E},\bar{\rho })$ of $(E,\rho )$ is compact. In general $\Psi$ is only ${\mathcal {E}}^u$-measurable and ${\mathcal {E}}^u=\mathcal {B}^u(\bar{E})|_E$. If X is a Borel right process, then $\Psi$ is ${\mathcal {E}}$-measurable.

The topology on E induced by the metric $\rho$ is called the Ray topology on E. Actually it does not depend on the choice of d or ${\mathcal {C}}$, and in general, is not compatible to the original topology on E. Let $C_\rho (E)$ denote the space of $\rho$-uniformly continuous functions on E. Then for all $\alpha >0$, $U^\alpha (C_\rho (E))\subset C_\rho (E)$, $U^\alpha (C_d(E))\subset C_\rho (E)$ and $\mathcal {R}-\mathcal {R}$ is uniformly dense in $C_\rho (E)$; see [16, (17.8)]. Let ${\mathcal {E}}^r:=\sigma \{C_\rho (E)\}$, i.e. the $\sigma$-algebra on E generated by the Ray topology. Then ${\mathcal {E}}\subset {\mathcal {E}}^r\subset {\mathcal {E}}^u$ and $P_t(b{\mathcal {E}}^r)\subset b{\mathcal {E}}^r$, $U^\alpha (b{\mathcal {E}}^r)\subset b{\mathcal {E}}^r$ for $\alpha >0$.

We may now construct a resolvent $\bar{U}^\alpha$ on $\bar{E}$ by continuity that extends $U^\alpha$ on E. Let $\bar{f}\in C(\bar{E})$ be the continuous extension of $f\in C_\rho (E)$. Then $U^\alpha f\in C_\rho (E)$ and so $U^\alpha f$ extends continuously to $\overline{U^\alpha f}\in C(\bar{E})$. Define the map $\bar{U}^\alpha : C(\bar{E})\rightarrow C(\bar{E})$ as

$$\begin{aligned} \bar{U}^\alpha \bar{f}:=\overline{U^\alpha f},\quad \bar{f}\in C(\bar{E}). \end{aligned}$$

Then $(\bar{U}^\alpha )_{\alpha >0}$ is a so-called Ray resolvent on $C(\bar{E})$ in the sense of, e.g., [16, (9.4)]. The collection $(\bar{E},\bar{\rho },\bar{U}^\alpha )$ is called the Ray-Knight compactification (or Ray-Knight completion) of $(E,d,U^\alpha )$. It depends not only on E, d and $U^\alpha$ but also on the choice of ${\mathcal {C}}$. For every $x\in E$, $\bar{U}^\alpha (x,\cdot )$ is carried by $E\in \mathcal {B}^u(\bar{E})$ and its restriction to E is equal to $U^\alpha (x,\cdot )$. Let $\bar{P}_t$ be the Ray semigroup associated with $\bar{U}^\alpha$ on $\bar{E}$. Then for all $x\in E$ and $t\ge 0$, $\bar{P}_t(x,\cdot )$ is also carried by E and its restriction to E is equal to $P_t(x,\cdot )$.

The Ray process $\bar{X}$ associated with $\bar{U}^\alpha$ on $\bar{E}$ admits branching points $B:=\{x\in \bar{E}: \bar{P}_0(x,\cdot )\ne \delta _x\}$, but may lead to a certain right processes by restriction. Note that $B\in \mathcal {B}(\bar{E})$ and the set of non-branching points $D:=\bar{E}\setminus B$ always contains E. The (first) restriction of $\bar{X}$ to D ($\supset E$) is a Borel right process. On the other hand, put

$$\begin{aligned} E_R:=\{x\in \bar{E}: \bar{U}^\alpha (x,\cdot )\text { is carried by }E\}, \end{aligned}$$

which inherits the subspace topology from $\bar{E}$. Then $E_R$ is independent of $\alpha$, $E\subset E_R$, and $E_R$ is a Radon topological space. This space, called the Ray space, is also independent of d and ${\mathcal {C}}$, and depends only on $U^\alpha$ and the original topology on E. Furthermore, the (second) restriction of $\bar{X}$ to $E_D:=E_R\cap D$ ($\supset E$) is also a (not necessarily Borel) right process.