Path continuity of Markov processes and locality of Kolmogorov operators

Abstract

We prove that if we are given a generator of a right Markov process with càdlàg paths and an open domain G in the state space, on which the generator has the local property expressed in a suitable way on a class ${\mathcal {C}}$ of test functions that is sufficiently rich, then the Markov process has continuous paths when it passes through G. The result holds for any Markov process which is associated with the generator merely on ${\mathcal {C}}$. This points out that the path continuity of the process is an a priori property encrypted by the generator acting on enough test functions, and this property can be easily checked in many situations. The approach uses potential theoretic tools and covers Markov processes associated with (possibly time-dependent) second order integro-differential operators (e.g., through the martingale problem) defined on domains in Hilbert spaces or on spaces of measures.

Appendix: Basics on right processes with càdlàg trajectories

Excessive functions, natural topologies and right processes Here we follow mainly the terminology of Beznea and Boboc [5], and we refer to the classical works [17, 33] and the references therein.

Let $(E, {\mathcal {B}})$ be a Lusin measurable space and ${\mathcal {B}}^u$ denote the corresponding universal $\sigma$-algebra. We denote by $(b)p{\mathcal {B}}$ the set of all numerical, (bounded) positive ${\mathcal {B}}$-measurable functions on E. Throughout, by ${\mathcal {U}}=(U_\alpha )_{\alpha >0}$ we denote a resolvent family of (sub-)Markovian kernels on $(E, {\mathcal {B}})$. If $q >0$, we set ${\mathcal {U}}_q:=(U_{q+\alpha })_{\alpha >0}$.

Definition 6.1

A ${\mathcal {B}}^u$-measurable function $v:E\rightarrow \overline{{\mathbb {R}}}_+$ is called excessive w.r.t. ${\mathcal {U}}$ (or ${\mathcal {U}}$-excessive) if $\alpha U_\alpha v \le v$ for all $\alpha >0$ and $\mathop {\sup }\nolimits _\alpha \alpha U_\alpha v =v$ pointwise; by $\mathcal {E({\mathcal {U}})}$ we denote the convex cone of all ${\mathcal {B}}$-measurable excessive functions w.r.t. ${\mathcal {U}}$.

If a ${\mathcal {B}}^u$-measurable function $w: E \rightarrow \overline{{\mathbb {R}}}_+$ is merely ${\mathcal {U}}_q$-supermedian (i.e. $\alpha U_{q+\alpha } w \le w$ for all $\alpha > 0$), then its ${\mathcal {U}}_q$-excessive regularization ${\widehat{w}}$ is defined as ${\widehat{w}}:=\mathop {\sup }\nolimits _\alpha \alpha U_{q +\alpha }w$.

(H) Throughout this paragraph we assume that ${\mathcal {E}}({\mathcal {U}}_q)$ is min-stable, contains the constant functions, and generates ${\mathcal {B}}$ for one (hence all) $q >0$.

Recall that (H) is necessary (yet not sufficient) for ${\mathcal {U}}$ to be associated to a right process, as defined below; a practical way to check this condition for a given resolvent of kernels is given in e.g. [12], page 846, and it is similar to ${{\textbf {(H}}}_{{\textbf {0}}}{{\textbf {)}}}$ from the beginning of Sect. 2.

Definition 6.2

• (i) The fine topology on E (associated with ${\mathcal {U}}$) is the coarsest topology on E such that every ${\mathcal {U}}_q$-excessive function is continuous for some (hence all) $q>0$.

• (ii) A topology $\tau$ on E is called natural if it is a Lusin topology (i.e. $(E,\tau )$ is homeomorphic to a Borel subset of a compact metrizable space) which is coarser than the fine topology, and whose Borel $\sigma$-algebra is ${\mathcal {B}}$.

Remark 6.3

The necessity of considering natural topologies comes from the fact that, in general, the fine topology is neither metrizable, nor countably generated.

There is a convenient class of natural topologies to work with (as we do in Sect. 2), especially when the aim is to construct a right process associated to ${\mathcal {U}}$ (see Definition 6.8). These topologies are called Ray topologies, and are defined as follows.

Definition 6.4

• (i) If $q >0$ then a Ray cone associated with ${\mathcal {U}}_q$ is a cone ${\mathcal {R}}$ of bounded ${\mathcal {U}}_q$-excessive functions which is separable in the supremum norm, min-stable, contains the constant function 1, generates ${\mathcal {B}}$, and such that $U_\alpha (({\mathcal {R}}-{\mathcal {R}})_+) \subset {\mathcal {R}}$ for all $\alpha > 0$.

• (ii) A Ray topology on E is a topology generated by a Ray cone.

Remark 6.5

• (i) Clearly, any Ray topology is a natural topology.

• (ii) By e.g. [12], Proposition 2.2, a Ray cone always exists (its existence is in fact equivalent with the validity of (H)) and may be constructed as follows: start with a countable subset ${\mathcal {A}}_0\subset p{\mathcal {B}}$ which separates the points of E, and define inductively

  $$\begin{aligned} {\mathcal {R}}_0&:=U_q ({\mathcal {A}}_0)\cup {\mathbb {Q}}_+\\ {\mathcal {R}}_{n+1}&:={\mathbb {Q}}_+\cdot {\mathcal {R}}_n \cup \left( \mathop {\sum }\limits _f{\mathcal {R}}_n \right) \cup \left( \mathop {\bigwedge }\limits _f {\mathcal {R}}_n\right) \cup \left( \mathop {\bigcup }\limits _{\alpha \in {\mathbb {Q}}_+}U_\alpha ({\mathcal {R}}_n)\right) \cup U_q (({\mathcal {R}}_n-{\mathcal {R}}_n)_+), \end{aligned}$$

  where by $\mathop {\sum }\nolimits _f {\mathcal {R}}_n$ resp. $\mathop {\bigwedge }\nolimits _f{\mathcal {R}}_n$ we denote the space of all finite sums (resp. infima) of elements from ${\mathcal {R}}_n$. Then, a Ray cone ${\mathcal {R}}$ is obtained by taking the closure of $\bigcup \nolimits _n {\mathcal {R}}_n$ w.r.t. the supremum norm.

The set of all natural topologies has the following remarkable structure:

Lemma 6.6

For any two natural topologies $\tau$ and $\tau '$ there exists a Ray (hence natural) topology which is finer than both $\tau$ and $\tau '$.

Proof

By Proposition 2.1 from Beznea and Boboc [6] if a natural topology is given then there exists a Ray topology which is finer than it. Therefore, we may assume that $\tau$ and $\tau '$ are Ray topologies induced by the Ray cones ${\mathcal {R}}$ and ${\mathcal {R}}'$ respectively and we may construct a Ray cone such that its closure in the supremum norm includes both ${\mathcal {R}}$ and ${\mathcal {R}}'$. $\square$

Further, following e.g. [17, Definition (3.1)], [30, Definition 1.5] or [5, Sect. A.2], let us make precise the (slightly different) notions for a Markov process that we are dealing with in this paper:

Definition 6.7

Let $(E,{\mathcal {B}})$ be a Lusin measurable space.

• (i) A collection $X:=(\Omega ^x, {\mathcal {F}}^x,{\mathcal {F}}^x_t, X_t^x, {\mathbb {P}}^x,x\in E_\Delta )$ is a simple (temporally homogeneous) Markov process with state space E, transition function $(P^\Delta _t)_{t\ge 0}$ and lifetime $\zeta ^x$ if

  • (i.1) $\Delta$ is an abstract isolated point of $E_\Delta :=E\cup \{\Delta \}$ with corresponding $\sigma$-algebra ${\mathcal {B}}_\Delta$ generated by ${\mathcal {B}}$,

  • (i.2) $(\Omega ^x,{\mathcal {F}}^x,({\mathcal {F}}^x_t)_{t\ge 0})$ is a filtered probability space and ${\mathbb {P}}^x$ is a probability on $(\Omega ^x,{\mathcal {F}}^x)$ for all $x\in E_\Delta$,

  • (i.3) For each $x\in E_\Delta$, $(X^x_t)_{t\ge 0}$ is an ${\mathcal {F}}^x_t$-adapted process with state space $E_\Delta$, and $\zeta ^x:\Omega \rightarrow [0,\infty ]$ is ${\mathcal {F}}^x$-measurable such that $X^x_t\in E, t<\zeta ^x$ and $X^x_t=\Delta , t\ge \zeta ^x$,

  • (i.4) $(P^\Delta _t)_{t\ge 0}$ is a semigroup of Markovian kernels on $(E_\Delta ,{\mathcal {B}}^u_\Delta )$ which is measurable, that is $[0,\infty )\times E_\Delta \ni (t,x)\mapsto P^\Delta _tf(x)\in {\mathbb {R}}$ is ${\mathcal {B}}([0,\infty ])\otimes {\mathcal {B}}^u_\Delta$-measurable for every $f\in b{\mathcal {B}}^u_\Delta$,

  • (i.5) $(X_t^\mu )_{t\ge 0}$ has the simple Markov property relative to $(P^\Delta _t)_{t\ge 0}$, namely

    $$\begin{aligned} {\mathbb {E}}^x\{f(X^x_{t+s} )|{\mathcal {F}}_s^x\}=P^\Delta _tf(X_s^x) \quad {\mathbb {P}}^x \text{-a.s. } \quad \forall s,t\ge 0, f\in b{\mathcal {B}}_\Delta , x\in E_\Delta . \end{aligned}$$

• (ii) We say that a simple Markov process X is normal if ${\mathbb {P}}^x\circ X(0)^{-1}=\delta _x, x\in E_\Delta$, and note that in this case

  $$\begin{aligned} P_t^\Delta f(x)={\mathbb {E}}^{x}\{f(X_t)\}, \quad t\ge 0, f\in b{\mathcal {B}}_\Delta , x\in E_\Delta , \end{aligned}$$

  (6.1)

  where ${\mathbb {E}}^x$ denotes the expectation under ${\mathbb {P}}^x$.

• iii) A collection $X=(\Omega , {\mathcal {F}}, {\mathcal {F}}_t, X_t, \theta _t, {\mathbb {P}}^x, x\in E_\Delta )$ is called a Markov process with state space E, transition function $(P_t^(\Delta )_{t\ge 0}$ shift operators $\theta _t, t\ge 0$, and lifetime $\zeta$ if:

  • (iii.1) $X=(\Omega , {\mathcal {F}}, {\mathcal {F}}_t, X_t, \theta _t, {\mathbb {P}}^x, x\in E_\Delta )$ is a simple normal Markov process with state space E and transition function $(P^\Delta _t)_{t\ge 0}$, that is the quadruplet $(\Omega , {\mathcal {F}}, {\mathcal {F}}_t, X_t, t\ge 0)=(\Omega ^x, {\mathcal {F}}^x,{\mathcal {F}}^x_t, X^x_t,t\ge 0)$ in i) does not depend on $x\in E_\Delta$,

  • (iii.2) For all $t,h\ge 0, \theta _t:\Omega \rightarrow \Omega$ is such that $X_t\circ \theta _h=X_{t+h}$.

Throughout, we set $f(\Delta )=0$ for all $f\in {\mathcal {B}}^u$, so that if $(P_t^\Delta )_{t\ge 0}$ is the transition function of a simple Markov process X with state space $E_\Delta$ then

$$\begin{aligned} P_tf(x):={\mathbb {E}}^x\{f(X^x_t)\}\quad \text{ for } \text{ all } f\in b{\mathcal {B}}^u, x\in E, \end{aligned}$$

renders a semigroup of sub-Markovian kernels (the transition function of X) on $(E,{\mathcal {B}}^u)$. We say that $(P_t)_{t\ge 0}$ is Borel if $P_t(b{\mathcal {B}})\subset b{\mathcal {B}}$; for simplicity, throughout the paper we shall deal with Borel semigroups if not otherwise stated.

Let ${\mathcal {U}}$ be the resolvent of X, i.e. for all $f\in b{\mathcal {B}}$ and $\alpha >0$

$$\begin{aligned} U_\alpha f(x)={\mathbb {E}}^{x}\left\{ \int _0^{\infty }e^{-\alpha t}f(X_t)dt\right\} ,\quad x\in E. \end{aligned}$$

To each probability measure $\mu$ on $(E, {\mathcal {B}})$ we associate the probability

$$\begin{aligned} {\mathbb {P}}^\mu (A):=\mathop {\int } {\mathbb {P}}^x(A)\; \mu (dx) \end{aligned}$$

for all $A \in {\mathcal {F}}$, and we consider the following enlarged filtration

$$\begin{aligned} \widetilde{{\mathcal {F}}}_t:= \bigcap \limits _\mu {\mathcal {F}}_t^\mu , \; \; \widetilde{{\mathcal {F}}}:= \bigcap \limits _\mu {\mathcal {F}}^\mu , \end{aligned}$$

where ${\mathcal {F}}^\mu$ is the completion of ${\mathcal {F}}$ under ${\mathbb {P}}^\mu$, and ${\mathcal {F}}_t^\mu$ is the completion of ${\mathcal {F}}_t$ in ${\mathcal {F}}^\mu$ w.r.t. ${\mathbb {P}}^\mu$.

Definition 6.8

A Markov process $X=(\Omega , {\mathcal {F}}, {\mathcal {F}}_t, X_t, \theta _t, {\mathbb {P}}^x, x\in E_\Delta )$ as in Definition 6.7(iii) is called right (Markov) process if the following additional hypotheses are satisfied:

• (i) The filtration $({\mathcal {F}}_t)_{t\ge 0}$ is right continuous and ${\mathcal {F}}_t=\widetilde{{\mathcal {F}}}_t, t\ge 0$.

• (ii) For one (hence all) $q >0$ and for each $f \in {\mathcal {E}}({\mathcal {U}}_q)$ the process f(X) has right continuous paths ${\mathbb {P}}^{x}$-a.s. for all $x\in E$.

• (iii) There exists a natural topology on E with respect to which the paths of X are ${\mathbb {P}}^{x}$-a.s. right continuous for all $x\in E$.

We would like to make the following convention: Whenever the space E is given along with a Lusin topology $\tau$ and there is no risk of confusion, by saying that X is a right process we implicitly assume that X has ${\mathbb {P}}^x$-a.s. $\tau$-right continuous paths for all $x\in E$.

Remark 6.9

Note that by [33, Definition 8.1 and Theorem 7.4], any right process has the strong Markov property, and this is a crucial ingredient for the proof of the main result of this paper, namely Theorem 3.6.

Also, recall that if ${\mathcal {U}}$ is the resolvent of a right Markov process X, then a non-negative real valued ${\mathcal {B}}$-measurable function v is excessive (w.r.t. ${\mathcal {U}}$) if and only if $(v(X_t))_{t\geqslant 0}$ is a right continuous ${\mathcal {F}}_t$ -supermartingale w.r.t. ${\mathbb {P}}^x$ for all $x\in E$; see e.g. [9, Proposition 1].

According to Blumenthal and Getoor [17, Chapter II, Theorem 4.8], or [33, Proposition 10.8 and Exercise 10.18], Definition 6.8 leads to a key probabilistic understanding of the fine topology, namely:

Theorem 6.10

If X is a right process, then an universally ${\mathcal {B}}$-measurable function f is finely continuous if and only if $(f(X_t))_{t\ge 0}$ has ${\mathbb {P}}^{x}$-a.s. right continuous paths for all $x\in E$. In particular, X has a.s. right continuous paths w.r.t. any natural topology on E.

Definition 6.11

If $q>0, u\in {\mathcal {E}}({\mathcal {U}}_q)$ and $A \in {\mathcal {B}}$, then the q-order reduced function of u on A is given by

$$\begin{aligned} R_q^A u = \inf \{ v \in {\mathcal {E}}({\mathcal {U}}_q): \, v\geqslant u \text{ on } A \}. \end{aligned}$$

$R_q^A u$ is merely supermedian w.r.t. ${\mathcal {U}}_q$, and we denote by $B_q^A u=\widehat{R_q^A u}$ its excessive regularization, called the balayage of u on A.

The following fundamental identification due to G.A. Hunt holds (see e.g. [19]):

Theorem 6.12

If X is a right process and $q>0$, then for all $u\in {\mathcal {E}}({\mathcal {U}}_q)$ and $A\in {\mathcal {B}}$

$$\begin{aligned} B_q^A u={\mathbb {E}}^x\{e^{-q T_A} u (X(T_A))\}, \end{aligned}$$

where $T_A:=\inf \{ t>0: X_t\in A \}$.

It is well known that $T_A$ is a stopping time and that $B^A_q u$ is universally measurable for all $A\in {\mathcal {B}}$ and $u\in b{\mathcal {B}}$; see [17] or [33].

Notions of “small” sets Assume that ${\mathcal {U}}$ is the resolvent of a right process X.

Definition 6.13

Let m be a $\sigma$-finite measure on E.

• (i) A set $A\in {\mathcal {B}}$ is called

  • “${\mathcal {U}}$-negligible” if $U_q (1_A)\equiv 0$ for one (hence all) $q> 0$.

  • “polar” if ${\mathbb {P}}^{x}(\{T_A<\infty \})=0$ for all $x\in E$.

  • “m-polar” if ${\mathbb {P}}^{x}(\{T_A<\infty \})=0$ m-a.e.

  • “m-inessential” provided that it is m-negligible and ${\mathbb {P}}^{x}(\{T_A<\infty \})=0$ for all $x\in E{\setminus } A$.

• (ii) A property is said to hold m-quasi-everywhere w.r.t. ${\mathcal {U}}$ (resp. ${\mathcal {U}}$-a.e.), if there exists an m-inessential (resp. a ${\mathcal {U}}$-negligible) set N s.t. the property holds for all $x\in E\setminus N$; on short, we write m-q.e. instead of m-quasi-everywhere.

Remark 6.14

For the reader convenience, let us recall several potential theoretic facts.

• (i) If $A \in {\mathcal {B}}$ is finely open and ${\mathcal {U}}$-negligible, then $A=\emptyset$.

• (ii) If m is a $\sigma$-finite measure on E s.t. $m(A)=0$ implies $U_1(1_A)=0$ m-a.e. for all $A\in {\mathcal {B}}$, then any finely open and m-negligible set $A\in {\mathcal {B}}$ is m-polar.

• (iii) Any m-inessential set is m-polar; conversely, it is known that any set which is m-polar and m-negligible is the subset of an m-inessential set.

• (iv) If $u\in b{\mathcal {B}}$ and v is ${\mathcal {B}}$-measurable s.t. $v=u$ q.e., then $B^A_{1}v$ is well defined and equal to $B^A_{1}u$ m-a.e.

Càdlàg paths in different topologies As far as we know, the stability of the right continuity of the paths under the change of the (natural) topology ensured by Theorem 6.10 can not be simply extended for left limits, without further conditions. To present such a condition, we adopt an $L^p$-framework, so let m be a $\sigma$-finite measure on E such that the resolvent ${\mathcal {U}}$ of X is strongly continuous on $L^{p}(m)$ for some $1\le p<\infty$.

Let us recall that an q-excessive function s is called regular if for every sequence of q-excessive functions $u_n\mathop {\nearrow } \nolimits _n u$ it holds that $R_q (u-u_n)\mathop {\searrow } \nolimits _n 0$, where $R_q$ is the reduction operator of level $q \ge 0$. A q-excessive function s is called m-regular if it has an m-version which is regular; see [4, 6] for more details.

Consider the following domination hypothesis:

${\mathbf {(D}}_{{\textbf{m}}}).$ There exists $0<f_0\in b{\mathcal {B}}$ such that for some $q >0$ every q-excessive function v dominated by $U_q f_0$ is m-regular.

The role of condition ${\mathbf {(D}}_{{\textbf{m}}}{\mathbf {)}}$ is expressed by the following fact which is a consequence of the results from Beznea and Boboc [6].

Proposition 6.15

If X is a right process and ${\mathbf {(D}}_{{\textbf{m}}}{\mathbf {)}}$ holds then X has ${\mathbb {P}}^x$-a.s. càdlàg trajectories in E on $[0, \zeta )$ m-q.e., with respect to all natural topologies.

Proof

Let $\tau$ be an arbitrary natural topology on E. By Lemma 6.6 from Appendix, we may replace $\tau$ with a finer Ray topology on E. Now, with condition ${\mathbf {(D}}_{{\textbf{m}}}{\mathbf {)}}$ in force, Theorems 1.5 and 1.3 from Beznea and Boboc [6] entail that X has càdlàg trajectories in E on $[0, \zeta )$ with respect to $\tau$, ${\mathbb {P}}^m$-a.e. Now, by the proof of Proposition 5.1 from [12] we have that the function

$$\begin{aligned} E\ni x\mapsto {\mathbb {P}}^x\left( \left\{ \omega \in \Omega : t\mapsto X_t(\omega ) \text{ is } \text{ not } \text{ c }\acute{\text {a}}\text{ dl }\acute{\text {a}}\text{ g } \text{ with } \text{ respect } \text{ to } \tau \right\} \right) \end{aligned}$$

is excessive, hence $\{v>0\}$ is finely open. Also, the above discussion leads to $m(\{v>0\})=0$, hence by Remark 6.14 we get that $\{v>0\}$ is contained in an m-inessential set. In other words, X has càdlàg trajectories with respect to $\tau$, ${\mathbb {P}}^x$-a.s. m-q.e. $\square$

Existence of a right process with a given resolvent ${\mathcal {U}}$. Without further conditions, the assumption (H) from the beginning of this section, although necessary, is not sufficient to ensure the existence of a right process associated with ${\mathcal {U}}$, but there is always a larger space on which such a process exists, and let us briefly recall its construction.

We denote by $Exc({\mathcal {U}}_q)$ the set of all ${\mathcal {U}}_q$-excessive measures: $\xi \in Exc({\mathcal {U}}_q)$ if and only if $\xi$ is a $\sigma$-finite measure on E and $\xi \circ \alpha U_{q+\alpha } \le \xi$ for all $\alpha >0$.

Definition 6.16

Let $q >0$.

• (i) The energy functional associated with ${\mathcal {U}}_q$ is $L^{q}: Exc({\mathcal {U}}_q)\times {\mathcal {E}}({\mathcal {U}}_q) \rightarrow \overline{{\mathbb {R}}}_+$ given by

  $$\begin{aligned} L^{q}(\xi ,v):=\sup \{\mu (v) \;: \; \mu \text{ is } \text{ a } \sigma \text{- } \text{ finite } \text{ measure, } \mu \circ U_q \le \xi \} \end{aligned}$$

• (ii) The saturation of E (with respect to ${\mathcal {U}}_q$) is the set $E_1$ of all extreme points of the set $\{\xi \in Exc({\mathcal {U}}_q)\;: \; L^{q}(\xi ,1)=1\}$.

The map $E\ni x \mapsto \delta _x \circ U_q \in Exc({\mathcal {U}}_q)$ is an embedding of E into $E_1$ and every ${\mathcal {U}}_q$-excessive function v has an extension ${\widetilde{v}}$ to $E_1$, defined as ${\widetilde{v}}(\xi ):=L^{q}(\xi ,v)$. The set $E_1$ is endowed with the $\sigma$-algebra ${\mathcal {B}}_1$ generated by the family $\{{\widetilde{v}}: \; v\in {\mathcal {E}}({\mathcal {U}}_q)\}$. In addition, as in [7], sections 1.1 and 1.2, there exists a unique resolvent of kernels ${\mathcal {U}}^{1}=(U^{1}_\alpha )_{\alpha >0}$ on $(E_1, {\mathcal {B}}_1)$ which is an extension of ${\mathcal {U}}$ in the sense that $U^1_\alpha (1_{E_1{\setminus } E})\equiv 0$ and $(U^1_\alpha f)|_E=U_\alpha (f|_E)$ for all $f\in b{\mathcal {B}}_1,\alpha >0$, and it satisfies the assumption (H) from the beginning of this section; more precisely, it is given by

$$\begin{aligned} U^{1}_\alpha f(\xi )=L^{q}(\xi , U_\alpha (f|_E)) \text{ for } \text{ all } f \in bp{\mathcal {B}}_1, \xi \in E_1, \alpha >0. \end{aligned}$$

(6.2)

Notice that $(E_1,{\mathcal {B}}_1)$ is a Lusin measurable space, the map $x \mapsto \delta _x \circ U_q$ identifies E with a subset of $E_1$, $E\in {\mathcal {B}}_1$ and ${\mathcal {B}}={\mathcal {B}}_1|_E$.

We end this section with the following key result, according to (2.3) from Beznea and Röckner [12], Sects. 1.7 and 1.8 in [5], Theorem 1.3 from Beznea et al. [7], and section 3 in [8]:

Theorem 6.17

Suppose that assumption (H) from the beginning of this section is satisfied. Then there exists a right process on the saturation $(E_1,{\mathcal {B}}_1)$, associated with ${\mathcal {U}}^{1}$. Moreover, the following assertions are equivalent:

• (i) There exists a right process on E associated with ${\mathcal {U}}$.

• (ii) The set $E_1\setminus E$ is polar (w.r.t. ${\mathcal {U}}_1$).

Existence of a right process with càdlàg trajectories associated with the resolvent ${\mathcal {U}}$ We end this section by recalling a general result that guarantees the existence of a right process with càdlàg trajectories associated to a given sub-Markovian resolvent of kernels ${\mathcal {U}}=(U_\alpha )_{\alpha >0}$ on a general Lusin measurable space $(E,{\mathcal {B}})$.

(Hypothesis). Assume that ${\mathcal {C}}\subset b{\mathcal {B}}$ such that

• (H1) ${\mathcal {C}}$ is a vector lattice, $1\in {\mathcal {C}}$.

• (H2) There exists a countable subset of ${\mathcal {C}}_+$ which separates the points of E.

• (H3) $U_\alpha ({\mathcal {C}})\subset {\mathcal {C}}$ for all $\alpha >0$.

• (H4) $\lim \nolimits _{\alpha \rightarrow \infty } \alpha U_\alpha f=f$ pointwise on E.

• (H5) There exists a ${\mathcal {B}}$-measurable function $v\in {\mathcal {E}}({\mathcal {U}}_q)$ for some $q \geqslant 0$ such that $v<\infty$ and $\{v\leqslant n\},n\geqslant 1$, is $\tau ({\mathcal {C}})$-compact, $n\geqslant 1$, is $\tau ({\mathcal {C}})$-compact; here $\tau ({\mathcal {C}})$ denotes the topology on E generated by ${\mathcal {C}}$.

Theorem 6.18

If (H1)–(H5) hold, then there exists a càdlàg right process X on E, endowed with the topology $\tau ({\mathcal {C}})$, with resolvent ${\mathcal {U}}$.

Proof

By Proposition 2.2 from Beznea and Röckner [12] there exists a Ray cone ${\mathcal {R}}$ such that the Ray topology generated by ${\mathcal {R}}$ is smaller than $\tau ({\mathcal {C}})$. Let $K_n=\{v\leqslant n\}$, $n\geqslant 1.$ It is an increasing sequence of $\tau ({\mathcal {C}})$-compact and by Theorem 6.12 if follows that $\lim _n T_n= + \infty$ ${\mathbb {P}}^x$-a.s. for every $x\in E$, where $T_n:= \inf \{ t>0: X_T \in E\setminus K_n \}.$ Notice that the Ray topology and $\tau ({\mathcal {C}})$ coincide on each compact set $K_n$. Applying Lemma 3.5 from Beznea et al. [8] and Theorem 6.17 it follows that there exists a right process X with state space E and ${\mathcal {U}}$ as associated resolvent. By Theorem 1.3 from Beznea and Boboc [6] we conclude now that X has càdlàg trajectories in the topology $\tau ({\mathcal {C}})$. $\square$

Remark 6.19

Results related to the domination hypothesis ($\mathbf{D_m}$) and to hypothesis (H5) may be found in [13, 14], in terms of the tightness property for the associated capacities; see also [29].