Quasimartingales associated to Markov processes

TL;DR

This paper characterizes functions of Markov processes whose evaluations are quasimartingales, showing they are differences of excessive functions, and explores their stability under various process transformations.

Contribution

It provides a complete characterization of quasimartingale functions for Markov processes and extends Fukushima's semimartingale results to semi-Dirichlet forms.

Findings

u(X) is a quasimartingale iff u is the difference of two finite excessive functions

Quasimartingale property is preserved under killing, time change, and subordination

Extension of semimartingale characterization to semi-Dirichlet forms

Abstract

For a fixed right process $X$ we investigate those functions $u$ for which $u(X)$ is a quasimartingale. We prove that $u(X)$ is a quasimartingale if and only if $u$ is the dif- ference of two finite excessive functions. In particular, we show that the quasimartingale nature of $u$ is preserved under killing, time change, or Bochner subordination. The study relies on an analytic reformulation of the quasimartingale property for $u(X)$ in terms of a certain variation of $u$ with respect to the transition function of the process. We provide sufficient conditions under which $u(X)$ is a quasimartingale, and finally, we extend to the case of semi-Dirichlet forms a semimartingale characterization of such functionals for symmetric Markov processes, due to Fukushima.

Equations

$$|\mathcal{E}(u,v)|\le c_n\Vert v{\Vert }_{\infty }\text{mbox}forallv\in {\mathcal{F}}_{b,F_n},$$

$$V(u):=\underset{\tau }{\sup }\{\sum _{i=1}^n{P}_{t_{i-1}}|u-P_{t_i-t_{i-1}}u|+P_{t_n}|u|\}$$

$$|\mathcal{E}(u,v)|\le c\Vert v{\Vert }_{\infty }\text{mbox}forallv\in {\mathcal{F}}_b$$

$${Var}^{\mathbb{P}}(Z):=\underset{\tau }{\sup }\mathbb{E}\{\sum _{i=1}^n|\mathbb{E}[Z_{t_i}-Z_{t_{i-1}}|{\mathcal{F}}_{t_{i-1}}]|+|Z_{t_n}|\}<\infty ,$$

$$u_1^{\tau }:=\sum _{i=1}^n{P}_{t_{i-1}}(u-P_{t_i-t_{i-1}}u{)}^{+}+P_{t_n}(u^{+})<\infty$$

$$u_2^{\tau }:=\sum _{i=1}^n{P}_{t_{i-1}}(u-P_{t_i-t_{i-1}}u{)}^{-}+P_{t_n}(u^{-})<\infty .$$

$$u_1:=\underset{n}{\sup }{u}_1^{{\tau }_n}=\underset{n}{\lim }{u}_1^{{\tau }_n}\mathrm{and}{u}_2:=\underset{n}{\sup }{u}_2^{{\tau }_n}=\underset{n}{\lim }{u}_2^{{\tau }_n}.$$

$$P_t{u}_2=\underset{n}{\sup }\underset{k}{\sup }\underset{l}{\sup }{P}_t{\phi }_{k,l}^n=\underset{n}{\sup }\underset{k}{\sup }\underset{l}{\sup }\underset{j}{\lim }{P}_{t_j}{\phi }_{k,l}^n\le \underset{j}{\mathrm{liminf}}{P}_{t_j}{u}_2\le u_2,$$

$$P_t{u}_1=\underset{k}{\sup }{P}_t(u_1\wedge k)=\underset{k}{\sup }\underset{j}{\lim }{P}_{t_j}(u_1\wedge k)\le u_1,$$

$$[V(u)<\infty ]\subset [\underset{n}{\sup }{V}_{{\tau }_n}(u)<\infty ]=A\subset [V(u^{+})+V(u^{-})<\infty ]\subset [V(u)<\infty ].$$

$$P_{M}^{\alpha}f(x):=\left\{\begin{array}[]{l}\mathbb{E}^{x}\int_{0}^{\infty}e^{-\alpha t}f(X_{t})dM_{t},\;x\in E_{M}\\[8.53581pt] f(x),\quad\quad x\in E\setminus E_{M},\end{array}\right.$$

$${Var}^{{\mathbb{P}}^x}(Mu(X))=V^{(Q_t)}u(x).$$

$${Var}^{{\mathbb{P}}^x}(Mu(X))=\underset{\tau }{\sup }{\mathbb{E}}^x\left\{\sum _{i=1}^n|{\mathbb{E}}^x[M_{t_{i-1}}u(X_{t_{i-1}})-M_{t_i}u(X_{t_i})|{\mathcal{F}}_{t_{i-1}}]|+M_{t_n}|u|(X_{t_n})\right\}$$

$$V^{(Q_{t}^{\alpha})}(u)=\left\{\begin{array}[]{l}V^{(Q_{t}^{\alpha}|_{E_{M}})}(u|_{E_{M}})(x),\;x\in E_{M}\\[8.53581pt] 0,\quad\quad x\in E\setminus E_{M}.\end{array}\right.$$

$${\tau }_t(\omega ):=\inf \{s:A_s(\omega )>t\},$$

$${\mathbb{E}}^x\{e^{-\alpha {\tau }_t}{u}_i(X_{{\tau }_t})\}\le u_i(x),x\in E,t\ge 0,i=\overline{1,2}.$$

$$Q_t{u}_i{|}_F(x)={\mathbb{E}}^x\{e^{-\alpha {\tau }_t}{u}_i(X_{{\tau }_t})\}\le u_i(x),x\in F,$$

$$P_t^{\mu }f:={\int }_0^{\infty }{P}_{s}f{\mu }_t(ds)\mathrm{for}\mathrm{all}f\in bp\mathcal{B},$$

$$U_{\alpha }(|u|+c)<\infty ,\underset{t\to \infty }{\mathrm{limsup}}{P}_t^{\alpha }|u|<\infty ,|P_{t}u-u|\le ct,t\ge 0,$$

$$|P_{t}u-u|\le ct,t\ge 0,\underset{t\in {\mathbb{R}}_{+}}{\sup }{P}_t^{\alpha }(|u|+c)=:b<\infty .$$

$$U_{\alpha }(|u|)(x_0)<\infty ,U_{\alpha }(|P_{t}u-u|)(x_0)\le const\cdot t,t\ge 0.$$

$${\tau }_n:=\left\{\frac{k}{2^n}:0\le k\le n\cdot 2^n\right\},n\ge 1.$$

$$|P_t^{\alpha }u-u|\le |P_{t}u|(1-e^{-\alpha t})+ct\le (const|P_{t}u|+c)t\le const(|u|+c)t$$

$$\underset{n}{\mathrm{limsup}}\left\{\sum _{k=1}^{n{2}^n-1}{P}_{\frac{k-1}{2^n}}^{\alpha }|P_{\frac{1}{2^n}}^{\alpha }u-u|\right\}\le const\cdot \underset{n}{\mathrm{limsup}}\{\frac{1}{2^n}\sum _{k=1}^{n{2}^n-1}{P}_{\frac{k-1}{2^n}}^{\alpha }(|u|+c)\}$$

$$\underset{n}{\lim }{V}_{{\tau }_n}^{\beta }(u)\le const\cdot \underset{n}{\mathrm{limsup}}\sum _{k=1}^{n{2}^n-1}{P}_{\frac{k-1}{2^n}}^{\beta }(|u|+c)\frac{1}{2^n}$$

$$D({\mathsf{L}}_p)=\{U_{\alpha }f|f\in L^p(\mu )\},$$

$${\mathsf{L}}_p(U_{\alpha }f)=\alpha U_{\alpha }f-f\mathrm{for}\mathrm{all}f\in L^p(\mu ),1\le p<\infty ,$$

$${\tilde{u}}_1=\underset{n}{\sup }{u}_1^{{\tau }_n}=\underset{n}{\sup }\left\{\sum _{i=1}^n{P}_{t_{i-1}}(u-P_{t_i-t_{i-1}}u{)}^{+}+P_{t_n}(u^{+})\right\},$$

$${\tilde{u}}_2=\underset{n}{\sup }{u}_2^{{\tau }_n}=\underset{n}{\sup }\left\{\sum _{i=1}^n{P}_{t_{i-1}}(u-P_{t_i-t_{i-1}}u{)}^{-}+P_{t_n}(u^{-})\right\},$$

$$P_t{\tilde{u}}_i=\underset{n}{\sup }{P}_t{u}_i^{{\tau }_n}=\underset{n}{\sup }\underset{k}{\lim }{P}_{t_k}{u}_i^{{\tau }_n}\le \underset{k}{\lim }{P}_{t_k}{\tilde{u}}_i\le {\tilde{u}}_i,$$

Full text

To appear in Transactions of the American Mathematical Society

**Quasimartingales associated to Markov processes

**

Lucian Beznea111Simion Stoilow Institute of Mathematics of the Romanian Academy, Research unit No. 2, P.O. Box 1-764, RO-014700 Bucharest, Romania, and University of Bucharest, Faculty of Mathematics and Computer Science (e-mail: lucian.beznea@imar.ro) and Iulian Cîmpean222Simion Stoilow Institute of Mathematics of the Romanian Academy, Research unit No. 2, P.O. Box 1-764, RO-014700 Bucharest, Romania, (e-mail: iulian.cimpean@imar.ro)

Abstract. For a fixed right process $X$ we investigate those functions $u$ for which $u(X)$ is a quasimartingale. We prove that $u(X)$ is a quasimartingale if and only if $u$ is the difference of two finite excessive functions. In particular, we show that the quasimartingale nature of $u$ is preserved under killing, time change, or Bochner subordination. The study relies on an analytic reformulation of the quasimartingale property for $u(X)$ in terms of a certain variation of $u$ with respect to the transition function of the process. We provide sufficient conditions under which $u(X)$ is a quasimartingale, and finally, we extend to the case of semi-Dirichlet forms a semimartingale characterization of such functionals for symmetric Markov processes, due to Fukushima.

Keywords. Semimartingale, quasimartingale, Markov process, excessive function, Dirichlet form, Fukushima decomposition, smooth measure.

Mathematics Subject Classification (2010). 60J45, 31C25, 60J40 (primary), 60J25, 60J35, 60J55, 60J57, 31C05 (secondary)

1 Introduction

Let us consider a (right) Markov process $X=(\Omega,\mathcal{F},\mathcal{F}_{t},X_{t},\mathbb{P}^{x})$ with state space $E$. In the celebrated paper [ÇiJaPrSh 80], the authors prove that a real-valued function $u$ on $E$ has the property that $u(X)$ is a semimartingale for each $\mathbb{P}^{x}$ if and only if there exists a sequence of finely open sets $(E_{n})_{n\geq 1}$ such that $\bigcup_{n}E_{n}=E$, the exit times $T_{n}$ of $E_{n}$ tend to infinity a.s., and $u$ is the difference of two $1$-excessive functions on each $E_{n}$. This characterization was later approached by Fukushima in [Fu 99] from a Dirichlet forms theory perspective. More precisely, he showed that if $X$ is associated with a symmetric Dirichlet form $(\mathcal{E},\mathcal{F})$ and $u\in\mathcal{F}$, then $\widetilde{u}(X)$ is a semimartingale if and only if there exist a nest $(F_{n})_{n\geq 1}$ and constants $(c_{n})_{n\geq 1}$ such that for each $n\geq 1$

[TABLE]

here $\widetilde{u}$ denotes a quasi-continuous version of $u$. The ideea of Fukushima in order to prove the sufficiency of inequality (1.1) was to assume first that $\mathcal{E}$ is a regular Dirichlet form so that, by Riesz representation, one has $\mathcal{E}(u,v)=\nu(v)$ for some Radon measure $\nu$ on $E$. The next step was to show that $\nu$ is a smooth measure, which means that the CAF from Fukushima decomposition is of bounded variation, hence $\widetilde{u}(X)$ is a semimartingale. The extension to quasi-regular symmetric Dirichlet forms was achieved via the so called ”transfer method”. This result was then used by the author in order to develop a deep stochastic counterpart of BV functions in both finite and infinite dimensions; beside the above mentioned paper, we refer the reader also to [Fu 00] and the references therein. As a matter of fact, the approach using Dirichlet forms dates back to the work of Bass and Hsu in [BaHs 90] where they showed that the reflected Brownian motion in a Lipschitz domain is a semimartingale, result which was later extended to (strong) Caccioppoli sets in [ChFiWi 93], where the authors investigate the quasimartingale structure of the process. It is worth to mention that in [ChFiWi 93] the authors consider quasimartingales only on finite intervals and not on the entire positive semi-axis, as we do (see Definition 2.1). Although it might seem a small difference, it is in fact the key point which makes our hole study achievable and, to the best of our knowledge, new.

The aim of this paper is twofold: firstly, we investigate those real-valued functions $u$ on $E$ for which $u(X)$ is a quasimartingale, and secondly, we study those functions $u$ for which $u(X)$ is a semimartingale by looking at their local quasimartingale structure. We briefly present below the structure and the main results of the paper:

In Section 2 we show that the quasimartingale property of $u(X)$ may be reformulated in terms of the variation

[TABLE]

of $u$ w.r.t. the semigroup $(P_{t})_{t\geq 0}$ of the process, which allows us to perform the study from a purely analytic point of view. The central results are Theorem 2.6 mainly saying that $\{x\in E\;:\;u(X)\;\mbox{is a quasimartingale w.r.t.}\;\mathbb{P}^{x}\}=\{V(u)<\infty\}$, and Corollary 2.7 according to which $u(X)$ is a quasimartingale (which by convention means for all $\mathbb{P}^{x},x\in E$) if and only if $u$ may be decomposed as the difference of two finite excessive functions. In particular, if the process is irreducible and $(e^{-\alpha t}u(X_{t}))_{t\geq 0}$ is a $\mathbb{P}^{x_{0}}$-quasimartingale for one $x_{0}\in E$, then it is a $\mathbb{P}^{x}$-quasimartingale for all $x\in E$. A Riesz type decomposition and some remarks on the space of differences of excessive functions are discussed in the end of the section.

In Section 3 we show that the quasimartingale property of functions is preserved under killing, time change, and Bochner subordination. In addition, we show that for a multiplicative functional $M$ with permanent points $E_{M}$, $(e^{-\alpha t}M_{t}u(X_{t}))_{t}$ is a quasimartingale if and only if $(e^{-\alpha t}u|_{E_{M}}(X^{M}))_{t}$ is a quasimartingale, where $X^{M}$ stands for the killed process by $M$; see Corollary 3.3. Also, in Proposition 3.5 we show that if $(e^{-\alpha t}u(X_{t}))_{t}$ is a quasimartingale, then so is the process $(e^{-\alpha\tau_{t}}u(Y_{t}))_{t}$, where $\tau$ is the inverse of an additive functional of $X$ and $Y$ denotes the corresponding time change process.

In Section 4 we provide tractable conditions for $u$ such that $(e^{-\alpha t}u(X_{t}))_{t}$ is a quasimartingale. We distinguish two ways of considering such conditions, which we treat separately: the first one involves the resolvent $\mathcal{U}=(U_{\alpha})_{\alpha}$ of the process, while the second approach is performed in an $L^{p}(\mu)$-context, where $\mu$ is a $\sigma$-finite sub-invariant measure. On brief, the key point is to search for an estimate of the type $U_{\alpha}(|P_{t}u-u|)\lesssim t$ for the first approach, and of the type $\mu(|P_{t}u-u|f)\lesssim t\|f\|_{\infty}$ in the $L^{p}$-context, but we refer the reader to Propositions 4.1 and 4.2 for the precise statemens; see also Proposition 4.5 for a condition in terms of the dual generator on $L^{p}$-spaces.

In the last section we look at quasimartingale and semimartingale functionals from the Dirichlet form theory point of view. More precisely, if $(\mathcal{E},\mathcal{F})$ is a (non-symmetric) Dirichlet form, then for an element $u\in\mathcal{F}$, an inequality of the type

[TABLE]

ensures that $(e^{-\alpha t}\widetilde{u}(X))_{t}$ is a quasimartingale; see Theorem 5.2. As a matter of fact, we show that this is true under a more general situation, when $\|v\|_{\infty}$ in (1.2) is replaced by $\|v\|_{\infty}+\|v\|_{L^{2}(\mu)}$, cf. Theorem 5.1. Then, in Theorem 5.3 we extend the semimartingale characterization due to Fukushima mentioned in the beginning of the introduction, to non-symmetric Dirichlet forms. Furthermore, in Corollary 5.4 we consider the situation when $u$ is not necessarily in $\mathcal{F}$ (e.g. $u\in\mathcal{F}_{\rm loc}$), under the additional hypothesis that the form has the local property. At this point we would like to emphasize that in contrast with previous work, in order to prove the sufficiency of conditions (1.1) or (1.2) we do not use Fukushima decomposition or Revuz correspondence. Instead, we employ heavily the results of the previous sections, and in fact, this approach enables us to extend Theorem 5.3 to semi-Dirichlet forms without further conditions; we do this in Theorem 5.5.

The paper ends with a few remarks concerning situations when it is sufficient to check inequalities (1.1) or (1.2) for $v$ belonging to a proper subspace of $\mathcal{F}$, like cores or special standard cores.

2 Quasimartingales of Markov processes

Before considering Markov processes, let us recall some classic facts about quasimartingales defined on a general probability space.

Definition 2.1**.**

Let $(\Omega,\mathcal{F},\mathcal{F}_{t},\mathbb{P})$ be a filtered probability space satisfying the usual hypotheses. An $\mathcal{F}_{t}$-adapted, right-continuous integrable process $(Z_{t})_{t\geq 0}$ is called $\mathbb{P}$-quasimartingale if

[TABLE]

where the supremum is taken over all partitions $\tau:0=t_{0}\leq t_{1}\leq\ldots\leq t_{n}<\infty$.

A classic result is Rao’s theorem according to which any quasimartingale has a unique decomposition as a sum between a local martingale and a predictable process with paths of locally integrable variation. In fact, the following characterization inspired our work (see e.g. [Pr 05], page 117):

Theorem 2.2**.**

(Rao) Let $(\Omega,\mathcal{F},\mathcal{F}_{t},\mathbb{P})$ as in Definition 2.1. A real-valued process is a $\mathbb{P}$-quasimartingale if and only if it is the difference of two positive right-continuous $\mathcal{F}_{t}$-adapted supermartingales.

Conversely, one can show that any semimartingale with bounded jumps is locally a quasimartingale.

Hereinafter we consider a right Markov process $X=(\Omega,\mathcal{F},\mathcal{F}_{t},X_{t},\mathbb{P}^{x})$ with state space a Lusin topological space $E$ endowed with the Borel $\sigma$-algebra $\mathcal{B}$, transition function $(P_{t})_{t\geq 0}$ and resolvent $\mathcal{U}=(U_{\alpha})_{\alpha>0}$. If $X$ has lifetime $\xi$ and cemetry point $\Delta$, we make the convention $u(\Delta)=0$ for all functions $u:E\to[-\infty,+\infty]$.

The aim of this section is to study those functions $u:E\to\mathbb{R}$ for which $u(X)$ is a $\mathbb{P}^{x}$-quasimartingale for all $x\in E$.

Definition 2.3**.**

Let $\alpha\geq 0$. A real valued $\mathcal{B}$-measurable function $u$ is called $\alpha$-quasimartingale function for $X$ if $(e^{-\alpha t}u(X_{t}))_{t\geq 0}$ is a $\mathbb{P}^{x}$-quasimartingale for all $x\in E$. When $\alpha=0$ we shall drop the index from notations.

Remark 2.4**.**

If $u$ is a quasimartingale function then, $\mathop{\sup}\limits_{t}P_{t}|u|(x)=\mathop{\sup}\limits_{t}\mathbb{E}^{x}|u(X_{t})|\leq{Var}^{\mathbb{P}^{x}}(u(X))<\infty,\;x\in E.$ Also, by the $\mathbb{P}^{x}$-a.s. right continuity of the trajectories $t\mapsto u(X_{t})$, $u$ must be finely continuous; see [BlGe 68], Theorem 4.8.

Notations. For a real valued function $u$ and a partition $\tau$ of $\mathbb{R}^{+}$, $\tau:0=t_{0}\leq t_{1}\leq\ldots\leq t_{n}<\infty$, we set

$V_{\tau}^{(P_{t})}(u):=\mathop{\sum}\limits_{i=1}^{n}P_{t_{i-1}}|u-P_{t_{i}-t_{i-1}}u|+P_{t_{n}}|u|$,

$V^{(P_{t})}(u):=\mathop{\sup}\limits_{\tau}V_{\tau}^{(P_{t})}(u)$,

where the supremum is taken over all finite partitions of $\mathbb{R}_{+}$. If there is no risk of confusion we shall write $V_{\tau}(u)$ and $V(u)$ instead of $V_{\tau}^{(P_{t})}(u)$ resp. $V^{(P_{t})}(u)$. Also, for $\alpha>0$ we set $V_{\tau}^{\alpha}(u):=V_{\tau}^{(P_{t}^{\alpha})}(u)$ and $V^{\alpha}(u):=V^{(P_{t}^{\alpha})}(u)$, where $P_{t}^{\alpha}:=e^{-\alpha t}P_{t},\alpha>0$.

Recall that for $\alpha\geq 0$, a $\mathcal{B}$-measurable function $f:E\rightarrow[0,\infty]$ is called $\alpha$-supermedian if $P_{t}^{\alpha}f\leq f$, $t\geq 0$. If $f$ is $\alpha$-supermedian and $\lim\limits_{t\to 0}P_{t}^{\alpha}f=f$ then it is called $\alpha$-excessive. The convex cone of all $\alpha$-supermedian (resp. $\alpha$-excessive) functions is denoted by $S(\mathcal{U}^{\alpha})$ (resp. $E(\mathcal{U}^{\alpha})$). If $\alpha=0$ we shall drop the index $\alpha$ from notations.

A set $A\in\mathcal{B}$ is called absorbing if $R_{\alpha}^{E\setminus A}1=0$ on $A$, where $R_{\alpha}^{A}1:=inf\{s\in E(\mathcal{U}^{\alpha}):s\geq 1_{A}\}$. We recall that if $A$ is absorbing then it is finely open, $U_{\alpha}1_{E\setminus A}=0$ on $A$, and the restriction of $X$ to $A$ is again a right process; see e.g. [Sha 88] or [BeRö 11]. Standard examples of absorbing sets are $[v=0]$ and $[v<\infty]$ for some $v\in E(\mathcal{U}^{\alpha})$ and $\alpha\geq 0$.

Definition 2.5**.**

A sequence $(\tau_{n})_{n\geq 1}$ of finite partitions of $\mathbb{R}_{+}$ is called admissible if it is increasing, $\mathop{\bigcup}\limits_{k\geq 1}\tau_{k}$ is dense in $\mathbb{R}_{+}$, and if $r\in\mathop{\bigcup}\limits_{k\geq 1}\tau_{k}$ then $r+\tau_{n}\subset\mathop{\bigcup}\limits_{k\geq 1}\tau_{k}$ for all $n\geq 1$.

The next theorem and its first corollary are the main results of this section.

Theorem 2.6**.**

Let $u$ be a real valued $\mathcal{B}$-measurable function such that $P_{t}|u|<\infty$ for all $t$. Then the following assertions hold.

i) ${Var}^{\mathbb{P}^{x}}(u(X))=V(u)(x),\;x\in E$.

ii) If $u_{1},u_{2}\in S(\mathcal{U})$ s.t. $u=u_{1}-u_{2}$ on the set $[u_{1}+u_{2}<\infty]$ then $V(u)\leq u_{1}+u_{2}$ on $[u_{1}+u_{2}<\infty]$.

iii) If $u$ is finely continuous, then there exist $u_{1},u_{2}\in E(\mathcal{U})$ such that $[V(u)<\infty]=[u_{1}+u_{2}<\infty]$ and $u=u_{1}-u_{2}$ on $[V(u)<\infty]$. In this case, the set $[V(u)<\infty]$ is absorbing and $[V(u)<\infty]=[\mathop{\sup}\limits_{n}V_{\tau_{n}}(u)<\infty]=[\lim\limits_{n}V_{\tau_{n}}(u)<\infty]$ for any admissible sequence of partitions $(\tau_{n})_{n}$.

One of the fundamental connections between potential theory and Markov processes is the relation between excessive functions and (right-continuous) supermartingales. More precisely, it is well known that for a non-negative real-valued measurable function $u$ we have that $u(X)$ is a supermartingale if and only if $u$ is excessive; see e.g. [LG 06], Proposition 13.7.1 and Theorem 14.7.1. The following essential consequence of Theorem 2.6 (and its proof), states that this connection may be extended between the space of differences of excessive function on the one hand, and quasimartingales on the other hand, in the same time revealing a Hahn-Jordan type decomposition.

Corollary 2.7**.**

A real valued $\mathcal{B}$-measurable function $u$ is a quasimartingale function for $X$ if and only if there exist two real-valued functions $u_{1},u_{2}\in E(\mathcal{U})$ such that $u=u_{1}-u_{2}$; in this case one can take $u_{1}:=\mathop{\sup}\limits_{n}V_{\tau_{n}}(u)$, where $(\tau_{n})_{n\geq 1}$ is any fixed sequence of admissible partitions of $\mathbb{R}_{+}$.

For the proof of Theorem 2.6 we need the following lemma. Because we found this result only as an exercise (left for the reader) in [Sha 88], Exercise 10.24 or [BlGe 68], Exercise 4.14, we include its complete proof below.

The first hitting time of a set $A\in\mathcal{B}$ by the process $X$ is defined by $T_{A}:=\inf\{t>0:X_{t}\in A\}$. It is well known that $T_{A}$ is a stopping time; see [BlGe 68] or [Sha 88].

Lemma 2.8**.**

If $u$ is finely continuous and bounded then so is $P_{s}u$ for all $s\geq 0$.

Proof.

Since $u$ is finely continuous, by [BlGe 68], Theorem 4.8, it follows that the mappings $t\mapsto u(X_{t})$ are right continuous a.s. Let $s>0$ and set $f:=P_{s}u$. In order to show that $f$ is finely continuous it is sufficient to prove that if $\varepsilon>0$ then $x$ is irregular for $A=f^{-1}([f(x)+\varepsilon,\infty))$ and $B=f^{-1}((-\infty,f(x)-\varepsilon])$. We treat only the first case. Let $(A_{n})_{n}$ be an increasing sequence of closed sets such that $T_{A_{n}}\searrow T_{A}\;\mathbb{P}^{x}$-a.s. By the zero-one law ([BlGe 68], Proposition 5.17), $\mathbb{P}^{x}(T_{A}=0)\in\{0,1\}$. Assume that $x$ is regular for $A$, i.e. $T_{A}=0$ $\mathbb{P}^{x}$-a.s. Then by the strong Markov property and dominated convergence theorem, $\mathbb{E}^{x}f(X_{T_{A_{n}}})=\mathbb{E}^{x}\{\mathbb{E}^{x}[u(X_{s+T_{A_{n}}})|\mathcal{F}_{T_{A_{n}}}]\}=\mathbb{E}^{x}u(X_{s+T_{A_{n}}})\mathop{\longrightarrow}\limits_{n}f(x)$. On the other hand, by the definition of $T_{A_{n}}$ we have that $f(X_{T_{A_{n}}})\geq f(x)+\varepsilon$, which contradicts the previous convergence. ∎

Proof of Theorem 2.6..

i). By the Markov property, for all $x\in E$

${Var}^{\mathbb{P}^{x}}(u(X))=\mathop{\sup}\limits_{\tau}\mathbb{E}^{x}\{\mathop{\sum}\limits_{i=1}^{n}|u(X_{t_{i-1}})-\mathbb{E}^{x}[u(X_{t_{i}})|\mathcal{F}_{t_{i-1}}]|+|u(X_{t_{n}})|\}$

$=\mathop{\sup}\limits_{\tau}\{\mathbb{E}^{x}\{\mathop{\sum}\limits_{i=1}^{n}|u(X_{t_{i-1}})-P_{t_{i}-t_{i-1}}u(X_{t_{i-1}})|+P_{t_{n}}|u|(x)\}\}$

$=\mathop{\sup}\limits_{\tau}\{\mathop{\sum}\limits_{i=1}^{n}\mathbb{E}^{x}[|u-P_{t_{i}-t_{i-1}}u|(X_{t_{i-1}})]+P_{t_{n}}|u|(x)\}$

$=\mathop{\sup}\limits_{\tau}\{\mathop{\sum}\limits_{i=1}^{n}P_{t_{i-1}}|u-P_{t_{i}-t_{i-1}}u|(x)+P_{t_{n}}|u|(x)\}=V(u)(x)$.

Note that the above expressions make sense because by hypothesis, $P_{t}|u|<\infty$ for all $t$.

ii). Since $u_{1},u_{2}\in S(\mathcal{U})$ we have that $A:=[u_{1}+u_{2}<\infty]$ satisfies $P_{t}1_{A^{c}}=\mathop{\lim}\limits_{n}P_{t}1_{[u_{1}+u_{2}>n]}\leq\mathop{\lim}\limits_{n}\dfrac{1}{n}P_{t}(1_{[u_{1}+u_{2}>n]}(u_{1}+u_{2}))\leq\mathop{\lim}\limits_{n}\dfrac{u_{1}+u_{2}}{n}=0$ on $A$ for all $t>0$. This leads to $1_{A}P_{t}f=1_{A}P_{t}(f1_{A})$ for all $\mathcal{B}$-measurable $f$ for which $P_{t}|f|<\infty$. Indeed, $|1_{A}P_{t}(f1_{A^{c}})|\leq 1_{A}P_{t}(|f|1_{A^{c}})=\mathop{\sup}\limits_{n}1_{A}P_{t}((|f|\wedge n)1_{A^{c}})\leq\mathop{\sup}\limits_{n}1_{A}nP_{t}1_{A^{c}}=0$.

By the previous remarks, we get

$\;\;1_{A}V(u)=\mathop{\sup}\limits_{\tau}\{\mathop{\sum}\limits_{i=1}^{n}1_{A}P_{t_{i-1}}|u-P_{t_{i}-t_{i-1}}u|+1_{A}P_{t_{n}}|u|\}$

$=\mathop{\sup}\limits_{\tau}\{\mathop{\sum}\limits_{i=1}^{n}1_{A}P_{t_{i-1}}|1_{A}u-1_{A}P_{t_{i}-t_{i-1}}(1_{A}u)|+1_{A}P_{t_{n}}1_{A}|u|\}$

$\leq\mathop{\sup}\limits_{\tau}\{\mathop{\sum}\limits_{i=1}^{n}1_{A}P_{t_{i-1}}|1_{A}u_{1}-1_{A}P_{t_{i}-t_{i-1}}(1_{A}u_{1})|+1_{A}P_{t_{n}}(1_{A}u_{1})\}$

$+\mathop{\sup}\limits_{\tau}\{\mathop{\sum}\limits_{i=1}^{n}1_{A}P_{t_{i-1}}|1_{A}u_{2}-1_{A}P_{t_{i}-t_{i-1}}(1_{A}u_{2})|+1_{A}P_{t_{n}}(1_{A}u_{2})\}$

$=1_{A}\mathop{\sup}\limits_{\tau}\{\mathop{\sum}\limits_{i=1}^{n}[P_{t_{i-1}}u_{1}-P_{t_{i}}u_{1}]+P_{t_{n}}u_{1}\}+1_{A}\mathop{\sup}\limits_{\tau}\{\mathop{\sum}\limits_{i=1}^{n}[P_{t_{i-1}}u_{2}-P_{t_{i}}u_{2}]+P_{t_{n}}u_{2}\}$

$=1_{A}(u_{1}+u_{2})$.

iii). For each partition $\tau:0=t_{0}\leq t_{1}\leq\ldots\leq t_{n}<\infty$, we set

[TABLE]

[TABLE]

Let $\prec$ denote the ordering of set containment and suppose that $\sigma$ and $\tau$ are two finite partitions of $\mathbb{R}_{+}$ s.t. $\sigma\prec\tau$. We claim that $u^{\sigma}_{i}\leq u^{\tau}_{i}$, $i=\overline{1,2}$. To see this, let $\sigma:0=t_{0}\leq t_{1}\leq\ldots\leq t_{n}<\infty$ and note that it is enough to consider $\tau$ as a partition obtained from $\sigma$ by adding an extra point $t$ before $t_{1}$, after $t_{n}$, or between some $t_{i}$ and $t_{i+1}$. In the first case we have $(u-P_{t_{1}}u)^{\pm}\leq(u-P_{t}u)^{\pm}+(P_{t}(u-P_{t_{1}-t}u))^{\pm}\leq(u-P_{t}u)^{\pm}+P_{t}(u-P_{t_{1}-t}u)^{\pm}$.

If $t\geq t_{n}$ then $P_{t_{n}}(u^{\pm})\leq P_{t_{n}}(u-P_{t-t_{n}}u)^{\pm}+P_{t}(u^{\pm})$, and if $t_{i}\leq t\leq t_{i+1}$, then $(u-P_{t_{i+1}-t_{i}}u)^{\pm}\leq(P_{t-t_{i}}(u-P_{t_{i+1}-t}u))^{\pm}+(u-P_{t-t_{i}}u)^{\pm}\leq P_{t-t_{i}}(u-P_{t_{i+1}-t}u)^{\pm}+(u-P_{t-t_{i}}u)^{\pm}$, hence $P_{t_{i}}(u-P_{t_{i+1}-t_{i}}u)^{\pm}\leq P_{t}(u-P_{t_{i+1}-t}u)^{\pm}+P_{t_{i}}(u-P_{t-t_{i}}u)^{\pm}$.

Therefore, $u^{\sigma}_{i}\leq u^{\tau}_{i}$, $i=\overline{1,2}$.

Let now $(\tau_{n})_{n\geq 1}$ be an admissible sequence of partitions of $\mathbb{R}_{+}$ and define

[TABLE]

Then $u_{1}+u_{2}=\mathop{\sup}\limits_{n}V_{\tau_{n}}(u)<\infty$ on $[V(u)<\infty]$.

Now, if $r\in\mathop{\bigcup}\limits_{n\geq 1}\tau_{n}$,

$P_{r}u_{1}=\mathop{\sup}\limits_{n}P_{r}u^{\tau_{n}}_{1}=\mathop{\sup}\limits_{n}\{\mathop{\sum}\limits_{i=1}^{n}P_{r+t_{i-1}}(u-P_{t_{i}-t_{i-1}}u)^{+}+P_{r+t_{n}}(u^{+})\}$

$=\mathop{\sup}\limits_{n}\{\mathop{\sum}\limits_{i=1}^{n}P_{r+t_{i-1}}(u-P_{r+t_{i}-(r+t_{i-1})}u)^{+}+P_{r+t_{n}}(u^{+})\}$

$\leq u_{1}$,

because the last supremum is taken over a class of partitions included in $\{\tau_{n}:n\geq 1\}$. Analogously, $P_{r}u_{2}\leq u_{2}$ for all $r\in\mathop{\bigcup}\limits_{n\geq 1}\tau_{n}$. Then,

$u_{2}+u=\mathop{\sup}\limits_{n}\left\{\mathop{\sum}\limits_{i=1}^{n}P_{t_{i-1}}(u-P_{t_{i}-t_{i-1}}u)^{-}+P_{t_{n}}(u^{-})+\mathop{\sum}\limits_{i=1}^{n}P_{t_{i-1}}(u-P_{t_{i}-t_{i-1}}u)+P_{t_{n}}u\right\}$

$=\mathop{\sup}\limits_{n}\left\{\mathop{\sum}\limits_{i=1}^{n}P_{t_{i-1}}(u-P_{t_{i}-t_{i-1}}u)^{-}+P_{t_{n}}(u^{-})\right\}$

$=u_{1}$,

and $u=u_{1}-u_{2}$ on $[\mathop{\sup}\limits_{n}V_{\tau_{n}}(n)<\infty]$.

Case 1. Assume that $u$ is lower bounded. We claim that $u_{1},u_{2}\in E(\mathcal{U})$ and $[V(u)<\infty]=[\mathop{\sup}\limits_{n}V_{\tau_{n}}(u)<\infty]$. First, note that if $u_{1},u_{2}\in E(\mathcal{U})$, since $u_{1}+u_{2}=\mathop{\sup}\limits_{n}V_{\tau_{n}}(u)$, $[V(u)<\infty]\subset[\mathop{\sup}\limits_{n}V_{\tau_{n}}(u)<\infty]$, and $u=u_{1}-u_{2}$ on $[u_{1}+u_{2}<\infty]$, by ii) we obtain $[V(u)<\infty]=[\mathop{\sup}\limits_{n}V_{\tau_{n}}(u)<\infty]$.

It remains to show that $u_{1},u_{2}\in E(\mathcal{U})$. By Lemma 2.8, the functions $\varphi_{k,l}^{n}:=\mathop{\sum}\limits_{i=1}^{n}P_{t_{i-1}}[(u-P_{t_{i}-t_{i-1}}(u\wedge k))^{-}\wedge l]$ are finely continuous and $u_{2}=\mathop{\sup}\limits_{n}\mathop{\sup}\limits_{k}\mathop{\sup}\limits_{l}\varphi_{k,l}^{n}$ is finely lower semi-continuous. Moreover, if $t\in\mathbb{R}_{+}$ and $(t_{j})_{j}\subset\mathop{\bigcup}\limits_{n\geq 1}\tau_{n}$, $t_{j}\searrow t$, then

[TABLE]

so $u_{2}$ is supermedian, and by [BeBo 04], Corollary 1.3.4, it is excessive. Now, since $u_{1}=u_{2}+u$ is finely continuous for $t\in\mathbb{R}_{+}$ and $(t_{j})_{j}$ as before,

[TABLE]

and $u_{1}\in E(\mathcal{U})$.

Case 2. Let now $u$ be arbitrary. Then $u^{+}=u_{1}-u_{1}\wedge u_{2}$ and $u^{-}=u_{2}-u_{1}\wedge u_{2}$ are finely continuous and of course, lower bounded. Applying Case 1 to $u^{+}$ and $u^{-}$ we have that $u=u^{+}-u^{-}$ is the difference of two real-valued excessive functions on $[V(u^{+})<\infty]\cap[V(u^{-})<\infty]$. Let us show that $[V(u)<\infty]=[\mathop{\sup}\limits_{n}V_{\tau_{n}}(u)<\infty]=[V(u^{+})<\infty]\cap[V(u^{-})<\infty]$, which completes the proof. Arguing as in the proof of ii), one can check that $A=[u_{1}+u_{2}<\infty]=[\mathop{\sup}\limits_{n}V_{\tau_{n}}(u)<\infty]$ satisfies $P_{r}1_{A^{c}}=0$ on $A$ for all $r\in\mathop{\bigcup}\limits_{n\geq 1}\tau_{n}$, and further, $V(u^{\pm})=\mathop{\sup}\limits_{n}V_{\tau_{n}}(u^{\pm})\leq u_{1}+u_{2}$ on $A$. Taking into account the sub-additivity of $f\mapsto V(f)$,

[TABLE]

∎

We say that the process $X$ is irreducible (in the strong sense) if the only non-empty absorbing set is the hole space $E$. Often in practice, the irreducibility of $\mathcal{U}$ is ensured by the strong Feller properly (i.e. $U_{\alpha}$ maps bounded measurable functions into continuous ones) in association with the topological irreducibility (i.e. $U_{\alpha}1_{D}>0$ for all open sets $D\subset E$); cf. e.g. [Ha 10].

Corollary 2.9**.**

Let $u$ be a real-valued $\mathcal{B}$-measurable finely continuous function and assume that there exists $x_{0}\in E$ such that $(e^{-\alpha t}u(X_{t}))_{t\geq 0}$ is a $\mathbb{P}^{x_{0}}$-quasimartingale for some $\alpha\geq 0$. The following assertions hold.

i) If $\mathcal{U}$ is irreducible then $(e^{-\alpha t}u(X_{t}))_{t\geq 0}$ is a $\mathbb{P}^{x}$-quasimartingale for all $x\in E$.

ii) If $\mathcal{U}$ is strong Feller and topologically irreducible then $\mathcal{U}$ is irreducible.

Proof.

i). By Proposition 3.1 below we have that $V^{\alpha}(u)(x_{0})=Var^{\mathbb{P}^{x_{0}}}((e^{-\alpha t}u(X_{t}))_{t\geq 0})<\infty$, hence $A:=[V^{\alpha}(u)<\infty]$ is absorbing (cf. Theorem 2.6, iii)) and non-empty. Since $\mathcal{U}$ is irreducible it follows that $A=E$.

ii). Let $B\in\mathcal{B}$ be absorbing and set $E_{0}:=[U_{1}1_{E\setminus B}=0]\supset B$. The strong Feller property implies that $E\setminus E_{0}$ is an open set and $1_{E\setminus B}\geq U_{1}1_{E\setminus B}\geq U_{1}1_{E\setminus E_{0}}$ leads to $E_{0}=E$. ∎

Following [Ge 80], $X$ is called recurrent if either $U1_{B}=0$ or $U1_{B}=\infty$ for all $B\in\mathcal{B}$. Getoor showed that $\mathcal{U}$ is recurrent if and only if any excessive function is constant, hence Corollary 2.7 gives the following quasimartingale characterization of recurrence.

Corollary 2.10**.**

$X$* is recurrent if and only if every quasimartingale function is constant.*

A Riesz type decomposition. Extending [Me 66] (see also [Sha 88], Chapter VI), a quasimartingale function $f$ is called (locally) harmonic if $f(X)$ is a $\mathbb{P}^{x}$-(local) martingale for all $x\in E$; it is called a potential function of class $(D)$ if for any sequence of stopping times $(T_{n})_{n}\nearrow\infty$, $\mathbb{E}^{x}[f(X_{T_{n}})]\to 0$.

Theorem 2.11**.**

If $u$ is a quasimartingale function for $X$, then $u$ may be decomposed as $u=h+v$, where $h$ is locally harmonic and $v$ is a potential function of class $(D)$.

Proof.

It follows by Corollary 2.7 and [Sha 88], Theorem (51.10). ∎

The space of differences of excessive functions. We saw that the space of $\alpha$-quasimartingale functions of $X$ is in fact the space of differences of real-valued $\alpha$-excessive functions. We end this section by collecting some useful observations on the dependence on $\alpha$ of the above mentioned spaces, in the same spirit as [BeLu 16], Remark 2.1.

Recall that $\mathcal{U}$ is called $m$-transient ($m$ is a fixed $\sigma$-finite sub-invariant measure for $\mathcal{U}$) if there exists $0<f\in L^{1}(m)$ such that $Uf<\infty$ $m$-a.e.

Proposition 2.12**.**

The following assertions hold.

i) For $\alpha,\beta\geq 0$, if $v\in E(\mathcal{U}_{\alpha})$ is real-valued such that $U_{\beta}v<\infty$, then $v$ is a difference of two real-valued $\beta$-excessive functions. In particular, $bE(\mathcal{U_{\alpha}})-bE(\mathcal{U}_{\alpha})$ is independent of $\alpha>0$.

ii) Let $m$ be a $\sigma$-finite sub-invariant measure for $\mathcal{U}$. Then:

ii.1) If $\alpha,\beta\geq 0$, $v\in E(\mathcal{U}_{\alpha})$ and $U_{\beta}v<\infty$ $m$-a.e. then $v$ is $m$-a.e. (hence q.e.) the difference of two $\beta$-excessive functions. In particular, the $L^{p}$-subspaces $L^{p}(m)\cap E(\mathcal{U}_{\alpha})-L^{p}(m)\cap E(\mathcal{U}_{\alpha})$ are independent of $\alpha>0$ for all $1\leq p\leq\infty$.

ii.2) If $\mathcal{U}$ is $m$-transient, then the $L^{1}$-subspaces $L^{1}(m)\cap E(\mathcal{U}_{\alpha})-L^{1}(m)\cap E(\mathcal{U}_{\alpha})$ are independent of $\alpha\geq 0$.

Proof.

i). Of course, we need to consider only the case $\beta<\alpha$. Let $w:=v+(\alpha-\beta)U_{\beta}v$. Then by hypothesis, $w<\infty$ and it is straightforward to check that $w$ is $\beta$-excessive. Hence $v=w-(\alpha-\beta)U_{\beta}v\in E(\mathcal{U}_{\beta})-E(\mathcal{U}_{\beta})$.

The proof of ii.1) is similar to the one for assertion i).

ii.2). By ii.1), it is sufficient to show that if $0\leq v\in L^{1}(m)$ then $Uv<\infty$ $m$-a.e. But this is true by a characterization of $m$-transience; see [BeCîRö 15]. ∎

3 Quasimartingale functions of transformed Markov processes

As in Section 2, $X=(\Omega,\mathcal{F},\mathcal{F}_{t},X_{t},\mathbb{P}^{x})$ is a right Markov process on $E$. Before we move on, we would like to remark that although in Section 1 we considered only $\mathcal{B}$-measurable functions, the results obtained there remain valid for functions measurable with respect to $\mathcal{B}^{u}$, the $\sigma$-algebra of all universally measurable sets in $E$.

Quasimartingales under killing. Let $M:=(M_{t})_{t\geq 0}$ be a right continuous decreasing multiplicative functionals $(MF)$ of $X$ and $E_{M}$ be the set of permanent points for $M$, $E_{M}:=\{x\in E:\mathbb{P}^{x}(M_{0}=1)=1\}$. As in [Sha 88], Proposition 56.5, define the kernels on $p\mathcal{B}^{u}$ by setting for $f\in p\mathcal{B}^{u}$, $\alpha\geq 0$, and $t\geq 0$

[TABLE]

$Q_{t}f(x):=\mathbb{E}^{x}\{f(X_{t})M_{t}\}$,

$W_{\alpha}f(x):=\mathbb{E}^{x}\int_{0}^{\infty}e^{-\alpha t}M_{t}f(X_{t})dt$.

It is well known that $(Q_{t})_{t}$ is a sub-Markovian semigroup of kernels on $(E,\mathcal{B}^{u})$ whose resolvent is $\mathcal{W}=(W_{\alpha})_{\alpha}\geq 0$.

Proposition 3.1**.**

Let $u$ be a real-valued $\mathcal{B}^{u}$-measurable function such that $P_{t}|u|<\infty$ for all $t\geq 0$. Then for all $x\in E$,

[TABLE]

Proof.

For $x\in E$,

[TABLE]

$=\mathop{\sup}\limits_{\tau}\mathbb{E}^{x}\left\{\mathop{\sum}\limits_{i=1}^{n}|M_{t_{i-1}}u(X_{t_{i-1}})-M_{t_{i-1}}Q_{t_{i}-t_{i-1}}u(X_{t_{i-1}})|+M_{t_{n}}|u|(X_{t_{n}})\right\}$

$=\mathop{\sup}\limits_{\tau}\left\{\mathop{\sum}\limits_{i=1}^{n}\mathbb{E}^{x}[M_{t_{i-1}}|u-Q_{t_{i}-t_{i-1}}u|(X_{t_{i-1}})]+Q_{t_{n}}|u|(x)\right\}$

$=\mathop{\sup}\limits_{\tau}\left\{\mathop{\sum}\limits_{i=1}^{n}Q_{t_{i-1}}|u-Q_{t_{i}-t_{i-1}}u|(x)+Q_{t_{n}}|u|(x)\right\}$

$=V^{(Q_{t})}u(x)$. ∎

Corollary 3.2**.**

Let $u$ be a real-valued $\mathcal{B}^{u}$-measurable function. If $\alpha\geq 0$, then $u$ is an $\alpha$-quasimartingale function if and only if it is the difference of two real-valued $\alpha$-excessive functions.

If $M$ is exact, then $E_{M}$ is finely open and the restriction $Q_{t}|_{E_{M}}$ of $(Q_{t})_{t\geq 0}$ to $E_{M}$ is the transition function of a right Markov process $(X_{t}^{M})_{t\geq 0}$ on $E_{M}$; see [Sha 88], Chapter VII.

Corollary 3.3**.**

Assume that $M$ is perfect. Then the following assertions hold.

i) Let $f$ be a real-valued $\mathcal{B}^{u}$-measurable function such that $U_{\alpha}|f|<\infty$ for some $\alpha\geq 0$ and set $u:=W_{\alpha}f$. Then $u$ is an $\alpha$-quasimartingale function for $X$.

ii) Let $u$ be a real-valued $\mathcal{B}^{u}$-measurable function, such that $Q_{t}|u|<\infty$ for all $t\geq 0$. Then for all $\alpha\geq 0$

[TABLE]

In particular, if $u$ is finely continuous then for all $\alpha\geq 0$, $(e^{-\alpha t}M_{t}u(X_{t}))_{t}$ is a $\mathbb{P}^{x}$-quasimartingale for all $x\in E$ if and only if $u|_{E_{M}}$ is an $\alpha$-quasimartingale function for $X^{M}$.

Proof.

i). Clearly, it is enough to consider $f\geq 0$. Then, the assertion follows since $u=U_{\alpha}f-P_{M}^{\alpha}U_{\alpha}f$ and $P_{M}^{\alpha}U_{\alpha}f\in E(\mathcal{U}^{\alpha})$; see e.g. [Sha 88], Proposition 56.5.

ii). The first assertion follows easily since $Q_{t}f\equiv 0$ on $E_{M}$ and $M_{t}\equiv 0$ $\mathbb{P}^{x}$-a.s. for $x\in E\setminus E_{M}$, while the second one is entailed by Proposition 3.1. ∎

Quasimartingales under time change. Let $A$ be a perfect continuous additive functional of $X$ ($AF$) and $F={supp}(A)$ its fine support. Then the inverse $\tau_{t}$ of $A_{t}$ defined

[TABLE]

is a stopping time for each $t\geq 0$ and the process $(\tau_{t})_{t\geq 0}$ is right continuous. Set $Y_{t}(\omega):=X_{\tau_{t}(\omega)}(\omega)$, $\mathcal{G}_{t}:=\mathcal{F}_{\tau_{t}}$, $t\geq 0$, $\mathcal{G}=\mathop{\bigcup}\limits_{t\geq 0}\mathcal{G}_{t}$. Then the process $Y=(\Omega,\mathcal{G},\mathcal{G}_{t},Y_{t},\mathbb{P}^{x})$ is a right process on $F$ and is called the time changed process of $X$ w.r.t. $A$; see [Sha 88], Chapter VII (more precisely, Theorem 65.9). We denote its resolvent by $\widehat{\mathcal{U}}$.

Corollary 3.4**.**

If $u$ is a quasimartingale function for $X$ then $u|_{F}$ is a quasimartingale function for $Y$. Conversely, if $F=E$, then any quasimartingale function for $Y$ is a quasimartingale function for $X$.

Proof.

If $u$ is a quasimartingale function for $X$, then by Corollary 2.7, $u=u_{1}-u_{2}$ with $u_{1},u_{2}\in E(\mathcal{U})$ and real-valued. But $E(\mathcal{U})|_{F}\subset E(\widehat{\mathcal{U}})$ (see [Sha 88], 65.12), so $u|_{F}$ is a quasimartingale function for $Y$ by the same Corollary 2.7. If $F=E$, the result follows by same arguments, since is this case, $E(\mathcal{U})=E(\widehat{\mathcal{U}})$; cf. [Sha 88], 65.13. ∎

The $\alpha$-quasimartingales are not preserved by time change, since $E(\mathcal{U}^{\alpha})\not\subset E(\overline{\mathcal{U}}^{\alpha})$, $\alpha>0$, in general. However, the following result holds.

Proposition 3.5**.**

If $u$ is an $\alpha$-quasimartingale function of $X$ for some $\alpha\geq 0$, then the process $(e^{-\alpha\tau_{t}}u(Y_{t}))_{t\geq 0}$ is a $\mathbb{P}^{x}$-quasimartingale w.r.t. the filtration $(\mathcal{G}_{t})_{t\geq 0}$ for all $x\in F$.

Proof.

If $u$ is an $\alpha$-quasimartingale function for $X$, then by Corollary 2.7, $u=u_{1}-u_{2}$ with $u_{1},u_{2}\in E(\mathcal{U}^{\alpha})$ finite on $E$. By Doob stopping theorem we have that

[TABLE]

On the other hand, $(\alpha\tau_{t})_{t\geq 0}$ is a perfect right-continuous $AF$ of $Y$, hence $(e^{-\alpha\tau_{t}})_{t\geq 0}$ is an exact and perfect $MF$ of $Y$; see [Sha 88], 54.11. Let $(Q_{t})_{t\geq 0}$ be the transition function of the process $Y$ killed by $(e^{-\alpha\tau_{t}})_{t\geq 0}$. Then

[TABLE]

which means that $u_{i}|_{F}$ is $(Q_{t})$-excessive, hence $V^{(Q_{t})}(u|_{F})<\infty$ (cf. Theorem 2.6, ii)). The result now follows since ${Var}^{\mathbb{P}^{x}}((e^{-\alpha\tau_{t}}(X_{\tau_{t}}))_{t\geq 0})=V^{(Q_{t})}(u|_{F})(x)$ by Proposition 2.1. ∎

Quasimartingales under Bochner subordination. Assume that $X$ is transient and let $\mu:=(\mu_{t})_{t\geq 0}$ be a vaguely continuous convolution semigroup of subprobability measures on $\mathbb{R}_{+}$. Define the subordinate $(P_{t}^{\mu})_{t\geq 0}$ of $(P_{t})_{t\geq 0}$ by

[TABLE]

whose resolvent is denoted by $\mathcal{U}^{\mu}:=(U_{\alpha}^{\mu})_{\alpha\geq 0}$. By [Lu 14], Theorem 3.3, $(P_{t}^{\mu})_{t\geq 0}$ is the transition function of a right process $X^{\mu}$ on $E$. Moreover, $E(\mathcal{U})\subset E(\mathcal{U}^{\mu})$, hence we have the following result.

Corollary 3.6**.**

Any quasimartingale function for $X$ is a quasimartingale function for $X^{\mu}$.

Example. Recall that a sub-Markovian resolvent of kernels $\mathcal{V}=(V_{\alpha})_{\alpha}$ is said to be $S$-subordinate to $\mathcal{U}$ if $E(\mathcal{U})\subset E(\mathcal{V})$; see [HmHm 09] and [Si 99].

By Corollary 2.7, it follows that the class of quasimartingale functions for $X$ is inherited by any right process whose resolvent is $S$-subordinate to $\mathcal{U}$. We remark that killing, time change, Bochner subordination, and any combination of them, may be regarded as $S$-subordinations w.r.t. $\mathcal{U}$, hence the quasimartingale functions for $X$ are preserved under such transformations. We emphasize that since the killing, time change, and Bochner subordination transformations do not commute in general, the order of any combination of them is relevant. We illustrate such a situation by looking at (Bochner) subordinate killed and killed subordinate Brownian motion. We follow [SoVo 03]; see also [HmJa 14], Example 7.

Let $(X_{t})_{t\geq 0}$ be a $d$-dimensional Brownian motion on $\mathbb{R}^{d}$ and $(\xi_{t})_{t\geq 0}$ an $\alpha$-stable subordinator starting at 0, $\alpha\in(0,1)$. Let $Y_{t}=X_{\xi_{t}}$ be the right process whose transition function is the subordinate $(P_{t}^{\mu})_{t\geq 0}$ of $(P_{t})_{t\geq 0}$ by means of the convolution semigroup $\mu$ induced by $(\xi_{t})_{t\geq 0}$. The generator of $Y$ is $-(-\Delta)^{\alpha}$, the fractional power of the negative Laplacian. Let now $D\subset\mathbb{R}^{d}$ be a domain and denote by $Y^{D}$ the killed upon leaving $D$, which is a right process obtained by killing $Y$ with the exact $MF$ $M_{t}=1_{[0,T_{D^{c}})}(t)$, $t\geq 0$, where $T_{D^{c}}(\omega):=\inf\{t>0\;|\;Y_{t}(\omega)\in D^{c}\}$.

Changing the order of transformations, let $Z$ be the right process obtained by first killing $X$ upon leaving $D$ and then subordinating the killed Brownian motion by means of $\mu$. The generator of $Z$ is $-(-\Delta|_{D})^{\alpha}$. As remarked in [HmJa 14], $Z$ is $S$-subordinate to $Y^{D}$, hence:

Corollary 3.7**.**

Any quasimartingale function for $Y^{D}$ is a quasimartingale function for $Z$.

4 Criteria for quasimartingale functions

In this section we present some sufficient conditions for a function to be an $\alpha$-quasimartingale function. In the first part we develop the study from the resolvent point of view, while in the last part we place ourselves in an $L^{p}$-context ($C_{0}$-semigroups and infinitesimal generators) with respect to a sub-invariant measure.

A resolvent approach. Again, we deal with a fixed right Markov process $X=(\Omega,\mathcal{F},\mathcal{F}_{t},X_{t},\mathbb{P}^{x})$ on $(E,\mathcal{B})$, with transition function $(P_{t})_{t\geq 0}$ and resolvent $\mathcal{U}=(U_{\alpha})_{\alpha>0}$.

The main result of this subsection is the following.

Proposition 4.1**.**

Let $u$ be a real-valued $\mathcal{B}$-measurable finely continuous function.

i) Assume there exist $\alpha\geq 0$ and $c\in p\mathcal{B}$ such that

[TABLE]

and the functions $t\mapsto P_{t}(|u|+c)(x)$ are Riemann integrable. Then $u$ is an $\alpha$-quasimartingale function.

ii) Assume there exist $\alpha\geq 0$, $c\in p\mathcal{B}$ such that

[TABLE]

Then $u$ is a $\beta$-quasimartingale function for all $\beta>\alpha$.

iii) Assume there exists $x_{0}\in E$ such that for some $\alpha\geq 0$

[TABLE]

Then $[V^{\beta}(u)<\infty]\neq\emptyset$ and if $\mathcal{U}$ is irreducible (e.g. strong Feller and topologically irreducible) then $u$ is a $\beta$-quasimartingale function for all $\beta>\alpha$.

Proof.

Assume that the conditions in i) are satisfied and let

[TABLE]

Clearly, $(\tau_{n})_{n\geq 1}$ is an admissible sequence of partitions of $\mathbb{R}_{+}$ (see Definition 2.5), hence, by Theorem 2.6, iii), we have to check that $\mathop{\lim}\limits_{n}V_{\tau_{n}}^{\alpha}(u)<\infty$. First, note that by hypotheses,

[TABLE]

for all $t$ small enough. Therefore,

$\mathop{\lim}\limits_{n}V_{\tau_{n}}^{\alpha}(u)=\mathop{\lim}\limits_{n}\left\{\mathop{\sum}\limits_{k=1}^{n2^{n}-1}P_{\frac{k-1}{2^{n}}}^{\alpha}|P_{\frac{1}{2^{n}}}^{\alpha}u-u|+P_{n2^{n}}^{\alpha}|u|\right\}$

$\leq\mathop{\lim\sup}\limits_{n}\left\{\mathop{\sum}\limits_{k=1}^{n2^{n}-1}P_{\frac{k-1}{2^{n}}}^{\alpha}|P_{\frac{1}{2^{n}}}^{\alpha}u-u|\right\}+\mathop{\lim\sup}\limits_{n}P_{n\cdot 2^{n}}^{\alpha}|u|$.

By hypothesis, $\mathop{\lim\sup}\limits_{n}P_{n\cdot 2^{n}}^{\alpha}|u|<\infty$. As for the other term, we have

[TABLE]

$={const}\cdot\int_{0}^{\infty}e^{-\alpha t}P_{t}(|u|+c)dt$

$={const}\cdot U_{\alpha}(|u|+c)<\infty$.

ii). Let $\beta>0$. Similarily to the above computations and noticing that $\lim\limits_{t\to\infty}P_{t}^{\beta}|u|=0$,

[TABLE]

$\leq const\cdot b\;\limsup\limits_{n}\sum\limits_{k=1}^{n2^{n}-1}e^{-(\beta-\alpha){\frac{k-1}{2^{n}}}}\frac{1}{2^{n}}$

$=const\cdot b\int\limits_{0}^{\infty}e^{-(\beta-\alpha)tdt}<\infty$.

iii). Let $\beta>\alpha$. Once we show that $[V^{\beta}(u)<\infty]\neq\emptyset$, the second assertion follows by Corollary 2.9. Taking into account Theorem 2.6, iii), we will show that $U_{\alpha}(\mathop{\lim}\limits_{n}V_{\tau_{n}}^{\beta}(u))(x_{0})<\infty$. Notice first that $\delta_{x_{0}}\circ U_{\alpha}$ is an $\alpha$-sub-invariant measure for $(P_{t})_{t}$, i.e. $U_{\alpha}(P_{t}^{\alpha}f)(x_{0})\leq U_{\alpha}f(x_{0})$ for all $f\in p\mathcal{B}$. Employing this property and using the hypotheses,

$U_{\alpha}(\mathop{\lim}\limits_{n}V_{\tau_{n}}^{\beta}(u))(x_{0})=\mathop{\lim}\limits_{n}U_{\alpha}(V_{\tau_{n}}^{\beta}(u))(x_{0})$

$=\mathop{\lim}\limits_{n}\left\{\mathop{\sum}\limits_{k=1}^{n\cdot 2^{n}-1}U_{\alpha}(P^{\beta}_{\frac{k-1}{2^{n}}}|u-P^{\beta}_{\frac{1}{2^{n}}}u|)(x_{0})+U_{\alpha}(P_{n}(|u|))(x_{0})\right\}$

$\leq\mathop{\lim\sup}\limits_{n}\mathop{\sum}\limits_{k=1}^{n2^{n}-1}e^{-(\beta-\alpha)\frac{k-1}{2^{n}}}U_{\alpha}|u-P^{\beta}_{\frac{1}{2^{n}}}u|(x_{0})$

$+\mathop{\lim\sup}\limits_{n}e^{-(\beta-\alpha)n}U_{\alpha}(|u|)(x_{0})$

$\leq{const}\;\cdot\int_{0}^{\infty}e^{-(\beta-\alpha)t}dt<\infty$. ∎

An $L^{p}$-approach. Throughout this subsection we assume that $\mu$ is a $\sigma$-finite sub-invariant measure for $(P_{t})_{t\geq 0}$. Hence $(P_{t})_{t\geq 0}$ and $\mathcal{U}$ extend to strongly continuous semigroup resp. resolvent family of contractions on $L^{p}(\mu)$, $1\leq p<\infty$. The corresponding generators $({\sf L}_{p},D({\sf L}_{p})\subset L^{p}(\mu))$ are defined by

[TABLE]

[TABLE]

with the remark that this definition si independent of $\alpha>0$.

The corresponding notations for the dual structure are $\widehat{P}_{t}$ and $(\widehat{\sf L}_{p},D(\widehat{\sf L}_{p}))$, and note that the adjoint of ${\sf L}_{p}$ is $\widehat{\sf L}_{p^{\ast}}$; $\frac{1}{p}+\frac{1}{p^{\ast}}=1$.

The main results of Section 2, namely Theorem 2.6 and its Corollary 2.7, can be reformulated in the $L^{p}(\mu)$ context. Although the proofs follow the same main ideas, they become simpler due to the strong continuity of $(P_{t})_{t\geq 0}$ on $L^{p}(\mu)$. Because we are mainly interested in the situation when $V(u)<\infty$ on $E$ except some negligible set, but also for simplicity, we present below the $L^{p}$-version of Corollary 2.7 only.

Proposition 4.2**.**

The following assertions are equivalent for a $\mathcal{B}$-measurable function $u\in\mathop{\bigcup}\limits_{1\leq p<\infty}L^{p}(\mu)$.

i) $u(X)$ is a $\mathbb{P}^{x}$-quasimartingale for $\mu$-a.e. $x\in E$.

ii) $V(u)<\infty$ $\mu$-a.e.

iii) For an admissible sequence of partitions of $(\tau_{n})_{n\geq 1}$ of $\mathbb{R}_{+}$, $\mathop{\sup}\limits_{n}V_{\tau_{n}}(u)<\infty$ $\mu$-a.e.

iv) There exist $u_{1},u_{2}\in E(\mathcal{U})$ finite $m$-a.e. such that $u=u_{1}-u_{2}$ $\mu$-a.e.

Proof.

We prove only iii) $\Rightarrow$ iv), just to point out the benefit of the strong continuity of $(P_{t})_{t\geq 0}$ on $L^{p}(\mu)$.

As in the proof of Theorem 2.6, iii), if we define $\widetilde{u}_{1}$ and $\widetilde{u}_{2}$ $\mu$-a.e. by

[TABLE]

[TABLE]

then $\widetilde{u}_{i}$ are finite $m$-a.e. and one can show that $P_{r}\widetilde{u}_{i}\leq\widetilde{u}_{i}$ for all $r\in\mathop{\bigcup}\limits_{n\geq 1}\tau_{n},i=\overline{1,2}$, and $u=\widetilde{u}_{1}-\widetilde{u}_{2}$ $\mu$-a.e.

If $t\in[0,\infty)$ and $\mathop{\bigcup}\limits_{n\geq 1}\tau_{n}\supset(t_{k})_{k}\searrow t$ then for $i=\overline{1,2}$ and $\mu$-a.e.,

[TABLE]

with the remark that the second holds $\mu$-a.e. because $u^{\tau_{n}}_{i}\in L^{p}(\mu)$ and $(P_{t})_{t\geq 0}$ is strongly continuous. Then, cf. e.g. [BeCîRö 15], Proposition 2.4, there exist two $\mathcal{B}$-measurable functions $u_{1},u_{2}\in E(\mathcal{U})$ s.t. $\widetilde{u}_{i}=u_{i}$ $\mu$-a.e., and finally, $u=u_{1}-u_{2}$ $\mu$-a.e. ∎

Remark 4.3**.**

i) We point out that for the proof of Proposition 4.2 we did not really used the fact that $\mu$ is sub-invariant, but just that $(P_{t})_{t}$ is strongly continuous on $L^{p}(\mu)$. In particular, Proposition 4.2 remains true for $u\in L^{\infty}(\mu)$ if we regard $(P_{t})_{t}$ as a strongly continuous semigroup on $L^{1}(\widehat{U}_{1}f\cdot\mu)$ for some $0<f\in L^{1}(\mu)$.

ii) If $u$ is a $\mathcal{B}$-measurable finely continuous function from $\mathop{\bigcup}\limits_{1\leq p\leq\infty}L^{p}(\mu)$ satisfying any of the equivalent assertions in Proposition 4.2, then the decomposition $u=u_{1}-u_{2}$ with $u_{1},u_{2}\in E(\mathcal{U})$ holds q.e.

Now, we focus our attention on a class of $\alpha$-quasimartingale functions which arises as a natural extension of $D({\sf L}_{p})$. First of all, it is clear that any function $u\in D({\sf L}_{p})$, $1\leq p<\infty$, has a representation $u=U_{\alpha}f=U_{\alpha}(f^{+})-U_{\alpha}(f^{-})$ with $U_{\alpha}(f^{\pm})\in E(\mathcal{U}^{\alpha})\cap L^{p}(\mu)$. In particular, $u$ has an $\alpha$-quasimartingale version for all $\alpha>0$. Moreover, $\|P_{t}u-u\|_{p}=\left\|\int_{0}^{t}P_{s}{\sf L}_{p}uds\right\|_{p}\leq t\|{\sf L}_{p}u\|_{p}$. Conversely, if $1<p<\infty$, $u\in L^{p}(\mu)$, and $\|P_{t}u-u\|_{p}\leq{const}\cdot t$, $t\geq 0$, then due to the reflexivity of $L^{p}$ we have that the family $\{\frac{P_{t}u-u}{t}\}_{t\geq 0}$ is weakly relatively compact, and by duality one can easily check that any weakly convergent subsequence $(\frac{P_{t_{n}}u-u}{t_{n}})_{t_{n}\to 0}$ has the same limit. Therefore $\frac{P_{t}u-u}{t}$ is weakly convergent to a limit from $L^{p}(\mu)$ as $t$ tends to [math], and by [Sa 99], Lemma 32.3, it is strongly convergent and $u\in D({\sf L}_{p})$. But this is no longer the case if $p=1$, and in general, $\|P_{t}u-u\|_{1}\leq{const}\cdot t$ does not imply $u\in D({\sf L}_{1})$. However, this last condition on $L^{1}(\mu)$ is still sufficient to guarantee that $u$ is an $\alpha$-quasimartingale function. In fact, the following general characterization holds.

Proposition 4.4**.**

Let $1\leq p<\infty$ and suppose $\mathcal{A}\subset\{u\in L^{p^{\ast}}_{+}(\mu):\|u\|_{p^{\ast}}\leq 1\}$, $\widehat{P}_{s}\mathcal{A}\subset\mathcal{A}$ for all $s\geq 0$, and $E=\mathop{\bigcup}\limits_{f\in\mathcal{A}}{\rm supp}(f)$ $\mu$-a.e. Then the following assertions are equivalent for $u\in L^{p}(\mu)$.

i) $\sup\limits_{f\in\mathcal{A}}\int_{E}|P_{t}u-u|fd\mu\leq{const}\cdot t$ for all $t\geq 0$.

ii) For every $\alpha>0$ there exist $u_{1},u_{2}\in E(\mathcal{U}^{\alpha})$ which satisfy i), $\mathop{\sup}\limits_{f\in\mathcal{A}}\int_{E}(u_{1}+u_{2})fd\mu<\infty$, and $u=u_{1}-u_{2}$ $\mu$-a.e.

Proof.

Since ii) $\Rightarrow$ i) is clear, let us prove the other implication. Assume that $u$ satisfies i). Then taking $\widehat{P}_{s}f$ instead of $f$ in condition i) we get for all $s,t\geq 0$

[TABLE]

$\leq\left[{const}\cdot t+(1-e^{-\alpha t})\|u\|_{p}\|f\|_{p^{\ast}}\right]e^{-\alpha s}$

$\leq{const}\cdot te^{-\alpha s}$.

Let now $\tau_{n}:=\left\{\dfrac{k}{2^{n}}:0\leq k\leq n2^{n}\right\}$, $n\geq 1$. Then, for $\alpha>0$

$\int_{E}\mathop{\sup}\limits_{n}V_{\tau_{n}}^{\alpha}(u)fd\mu=\mathop{\lim}\limits_{n}\mathop{\sum}\limits_{k=1}^{n2^{n}}\int_{E}P_{\frac{k-1}{2^{n}}}^{\alpha}|P_{\frac{1}{2^{n}}}^{\alpha}u-u|fd\mu$

$\leq{const}\cdot\mathop{\lim}\limits_{n}\mathop{\sum}\limits_{k=1}^{n2^{n}}e^{-\alpha\frac{k-1}{2^{n}}}\frac{1}{2^{n}}$

$={const}\cdot\int_{0}^{\infty}e^{-\alpha t}dt<\infty$

for all $f\in\mathcal{A}$. Hence $\mathop{\sup}\limits_{n}V_{\tau_{n}}^{\alpha}(u)<\infty$ $\mu$-a.e. and by Proposition 4.2 we have that $u=u_{1}-u_{2}$ $\mu$-a.e. with $u_{1},u_{2}\in E(\mathcal{U}^{\alpha})$. Moreover, inspecting the way $u_{1}$ and $u_{2}$ have been constructed, we have that $u_{1}+u_{2}=\mathop{\sup}\limits_{n}V_{\tau_{n}}^{\alpha}(u)$ $\mu$-a.e., hence $\mathop{\sup}\limits_{f\in\mathcal{A}}\int_{E}(u_{1}+u_{2})fd\mu<\infty$. Moreover, for $r\in\mathop{\bigcup}\limits_{n\geq 1}\tau_{n}$ and $i=\overline{1,2}$,

[TABLE]

$+\mathop{\sum}\limits_{i=1}^{n2^{n}}P_{r}^{\alpha}P_{\frac{i-1}{2^{n}}}^{\alpha}(u-P_{\frac{1}{2^{n}}}^{\alpha}u)^{\pm}+P_{r}^{\alpha}P_{n-r}^{\alpha}(u^{\pm})\}$

$=\mathop{\lim}\limits_{n}\left\{\mathop{\sum}\limits_{k=1}^{r2^{n}}P_{\frac{k-1}{2^{n}}}^{\alpha}(u-P_{\frac{1}{2^{n}}}^{\alpha}u)^{\pm}+P_{r}^{\alpha}(u-P_{\frac{1}{2^{n}}}^{\alpha}u)^{\pm}\right\}+P_{r}^{\alpha}u^{i}$.

Therefore

$\int_{E}|u_{i}-P_{r}^{\alpha}u_{i}|fd\mu\leq\mathop{\lim}\limits_{n}\mathop{\sum}\limits_{k=1}^{r2^{n}}\int_{E}P_{\frac{k-1}{2^{n}}}^{\alpha}|u-P_{\frac{1}{2^{n}}}^{\alpha}u|fd\mu$

$\leq{const}\cdot\int_{0}^{r}e^{-\alpha t}dt$

$={const}\cdot r$

for all $f\in\mathcal{A}$, $i=\overline{1,2}$, $r\in\mathop{\bigcup}\limits_{n\geq 1}\tau_{n}$, where the above constant is independent of $f\in\mathcal{A}$, $i=\overline{1,2}$, and $r\in\mathop{\bigcup}\limits_{n\geq 1}\tau_{n}$.

We claim that $\int_{E}(u_{i}-P_{t}^{\alpha}u_{i})fd\mu\leq{\rm const}\cdot t$ for all $t\geq 0$, $i=\overline{1,2}$, and $f\in\mathcal{A}$. Since the desired inequality holds for all $r\in\mathop{\bigcup}\limits_{n\geq 1}\tau_{n}$ and $0\leq u_{i}-P_{r}^{\alpha}u_{i}\leq u_{i}$, by dominated convergence it is sufficient to show that for each $f\in\mathcal{A}$, $P_{r_{k}}^{\alpha}u_{i}$ converges $f\cdot\mu$-a.e. on a subsequence to $P_{t}^{\alpha}u_{i}$, whenever $\mathop{\bigcup}\limits_{n\geq 1}\tau_{n}\ni r_{k}\mathop{\searrow}\limits_{k}t\geq 0$. To see this, let $\nu:=\widehat{U}_{\alpha}f\cdot\mu$ and note that $u_{i}\in L^{1}(\nu)$. Since $\nu$ is a sub-invariant measure for $(P_{t}^{\alpha})_{t\geq 0}$ we have that $(P_{t}^{\alpha})_{t\geq 0}$ is strongly continuous on $L^{1}(\nu)$, hence if $\mathop{\bigcup}\limits_{n\geq 1}\tau_{n}\ni r_{k}\searrow t\geq 0$ it follows that on a subsequence, $(P_{r_{k}}^{\alpha}u_{i})_{k\geq 1}$ converges $\nu$-a.e. to $P_{t}^{\alpha}u_{i}$. Since $f\cdot\mu\ll\nu$ we obtain that the above convergence holds $f\cdot\mu$-a.e. So,

[TABLE]

and finally

[TABLE]

for all $t\geq 0$, $i=\overline{1,2}$, and independently on $f\in\mathcal{A}$. ∎

We can interpert condition i) from Proposition 4.4 in terms of the adjoint generator as follows.

Proposition 4.5**.**

Let $p\in(1,\infty)$ and $q\in[1,\infty]$. The following assertions are equivalent for $u\in L^{p}(\mu)$.

i) $|\mu(u\;\widehat{\sf L}_{p^{\ast}}v)|\leq{const}\cdot(\|v\|_{\infty}+\|v\|_{q})$ for all $v\in D(\widehat{\sf L}_{p^{\ast}})$.

ii) $u$ satisfies i) from Proposition 4.4 for all $\mathcal{A}=\{v\in L^{p^{\ast}}(\mu):\|v\|_{\infty}+\|v\|_{q}\leq 1\}$.

Proof.

i) $\Rightarrow$ ii). Let $f\in L^{\infty}(\mu)\cap L^{q}(\mu)\cap L^{p^{\ast}}(\mu)$. For $t\geq 0$ let $w:=\dfrac{1}{t}{\rm sgn}(P_{t}u-u)f\in L^{p^{\ast}}(\mu)$ and $v:=\int_{0}^{t}\widehat{P}_{s}wds\in D(\widehat{\sf L}_{p^{\ast}})$. Then $\widehat{\sf L}_{p^{\ast}}v=\widehat{P}_{t}w-w$, $\|v\|_{\infty}+\|v\|_{q}\leq 2(\|f\|_{\infty}+\|f\|_{q})$, and

[TABLE]

$=\int_{E}u\;\widehat{\sf L}_{p^{\ast}}vd\mu\leq 2\cdot{const}\cdot(\|f\|_{\infty}+\|f\|_{q})$.

Therefore, $\int_{E}|P_{t}u-u|fd\mu\leq{const}\cdot t$ for all $t\geq 0$ and $f\in\mathcal{A}$.

ii) $\Rightarrow$ i). If $\int\limits_{E}|P_{t}u-u|fd\mu\leq const\cdot t(\|f\|_{\infty}+\|f\|_{q})$, then by replacing $f$ with ${\rm sgn}(P_{t}u-u)f$ we get

[TABLE]

Now, if $f\in D(\widehat{\sf L}_{p^{\ast}})$ then assertion i) follows by letting $t$ tend to [math]. ∎

Example: adding jumps to a Markov process. Assume that $X$ is a standard process and $\sf N$ is a Markov kernel on $E$. As before, $\mu$ is a $\sigma$-finite sub-invariant measure for $\mathcal{U}$. We assume further that $\mu\circ{\sf N}\leq\mu$. It is well known that there exists a second Markov process $Y$ on $E$ whose infinitesimal generator is given by ${\sf Q}:={\sf L}-1+{\sf N}$; $D({\sf L})=D({\sf Q})$; cf. [Ba 79] or [BeSt 94], Theorem 1.8; see also [Op 16] for more general perturbations with kernels for generators of Markov processes. Let $\mathcal{V}=(V_{\alpha})_{\alpha}$ denote the resolvent of $Y$. Then $V_{\alpha}=U_{\alpha+1}+U_{\alpha+1}{\sf N}V_{\alpha}$ and

[TABLE]

for all $f\in L^{1}_{+}(\mu)$, which means that $\mu$ is $\mathcal{V}$ - sub-invariant. Therefore, we can extend $\sf Q$ on $L^{p}(\mu)$, $1\leq p<\infty$ by ${\sf Q}_{p}:={\sf L}_{p}-1+{\sf N}_{p}$, $D({\sf Q}_{p})=D({\sf L}_{p})$, where ${\sf L}_{p}$ and ${\sf N}_{p}$ are the corresponding $L^{p}(\mu)$-extensions of ${\sf L}$ and ${\sf N}$.

Let $(S_{t})_{t\geq 0}$ be the transition function of $Y$. Since $\mu$ is $(S_{t})_{t\geq 0}$-sub-invariant we have that $(S_{t})_{t\geq 0}$ extends to a $C_{0}$-semigroup of contractions on $L^{p}(\mu)$, $1\leq p<\infty$, for which we keep the same notation.

Clearly, since $E(\mathcal{V}_{\alpha})\subset E(\mathcal{U}_{\alpha+1})$, we get by Corollary 2.7 that any $\alpha$-quasimartingale function for $Y$ is an $(\alpha+1)$-quasimartingale function for $X$. Also, as remarked in [BeLu 16], Proposition 4.5, the spaces of differences of bounded functions from $E(\mathcal{U}_{\alpha+1})$ and respectively from $E(\mathcal{V}_{\alpha})$ are the same. Next, we show that the class of quasimartingale functions which are produced by the estimate in Proposition 4.4, i) (and in Corollary 4.5, ii)) are the same for both $X$ and $Y$.

Corollary 4.6**.**

Let $1\leq p<\infty$, $u\in L^{p}(\mu)$, and $\mathcal{A}$ be a bounded subset in $L^{p^{\ast}}$. Then i) from Proposition 4.4 is satisfied w.r.t. $(P_{t})_{t\geq 0}$ if and only if it is satisfied w.r.t. $(S_{t})_{t\geq 0}$.

Proof.

The result follows easily since ${\sf Q}_{p}$ is a bounded perturbation of ${\sf L}_{p}$, and by e.g. [EnNa 99], Corollary 1.11, there exists a constant $c$ s.t. $\|P_{t}-S_{t}\|_{p}\leq t\cdot c$, $t\geq 0$. ∎

5 Applications to Dirichlet forms

Let $E$ be a Hausdorff topological space with Borel $\sigma$-algebra $\mathcal{B}$, $\mu$ be a $\sigma$-finite measure on $\mathcal{B}$, and $\mathcal{E}$ be a bilinear form on $L^{2}(\mu)$ with dense domain $\mathcal{F}$; $\mathcal{E}_{\alpha}(\cdot,\cdot)=\mathcal{E}(\cdot,\cdot)+\alpha(\cdot,\cdot)$, $\alpha>0$.

Recall that $(\mathcal{E},\mathcal{F})$ is called a coercive closed form if:

i) $(\mathcal{E},\mathcal{F})$ is positive definite and closed on $L^{2}(\mu)$.

ii) $\mathcal{E}$ satisfies the (weak) sector condition, i.e. there exists a constant $k$ s.t.

[TABLE]

The coercive closed form $(\mathcal{E},\mathcal{F})$ is called a Dirichlet form if $u^{+}\wedge 1\in\mathcal{F}$ and both

iii) $\mathcal{E}(u+u^{+}\wedge 1,u-u^{+}\wedge 1)\geq 0$

iv) $\mathcal{E}(u-u^{+}\wedge 1,u+u^{+}\wedge 1)\geq 0$

hold for all $u\in\mathcal{E}$. If only iii) is satisfied then $(\mathcal{E},\mathcal{F})$ is called a semi-Dirichlet form.

A bilinear form $(\mathcal{E},\mathcal{F})$ on $L^{2}(\mu)$ is called a lower-bounded (semi) Dirichlet form if there exists $\alpha>0$ s.t. $(\mathcal{E}_{\alpha},\mathcal{F})$ is a (semi) Dirichlet form. If $(\mathcal{E},\mathcal{F})$ is a coercive closed form, let $(P_{t})_{t\geq 0}$ be the $C_{0}$-semigroup of contractions on $L^{2}(m)$ associated to $\mathcal{E}$, whose dual is denoted by $(\widehat{P}_{t})_{t\geq 0}$. Recall that condition iii) (resp. iv)) is equivalent with the sub-Markov property for $(P_{t})_{t\geq 0}$ (resp. $(\widehat{P}_{t})_{t\geq 0}$); see [MaRö 92], I.4.4.

Adopting the notations from [Fu 99], for a closed set $F\subset E$ we set:

$\mathcal{F}_{F}=\{v\in\mathcal{F}\;:\;v=0\;m\mbox{-a.e. on}\;E\setminus F\}$,

$\mathcal{F}_{b,F}=\{v\in\mathcal{F}_{F}\;:\;v\in L^{\infty}(\mu)\}$.

An increasing sequence of closed sets $(F_{n})_{n\geq 1}$ is called an $\mathcal{E}$-nest if $\mathop{\bigcup}\limits_{n=1}^{\infty}\mathcal{F}_{F_{n}}$ is $\mathcal{E}_{1}$-dense in $\mathcal{F}$. An element $f\in\mathcal{F}$ is called $\mathcal{E}$-quasi-continuous if there exists a nest $(F_{n})_{n\geq 1}$ such that $f|_{F_{n}}$ is continuous for each $n\geq 1$.

A (semi) Dirichlet form $(\mathcal{E},\mathcal{F})$ on $L^{2}(\mu)$ is called quasi-regular if there exist an $\mathcal{E}$-nest consisting of compact sets, an $\mathcal{E}_{1}$-dense subset of $\mathcal{F}$ whose elements admit $\mathcal{E}$-quasi-continuous versions, and a countable family of $\mathcal{E}$-quasi-continuous elements from $\mathcal{F}$ which separates the points of $\mathop{\bigcup}\limits_{n=1}^{\infty}E_{n}$ for a certain $\mathcal{E}$-nest $(E_{n})_{n\geq 1}$. It is well known that the quasi-regularity property is a necessary and sufficient condition for a semi-Dirichlet form to be (properly) associated to a $\mu$-tight special standard process $X$ (i.e. the semigroup $(P_{t})_{t}$ of $(\mathcal{E},\mathcal{F})$ is generated by the transition function of $X$); see [MaOvRö 95] or [MaRö 92] for details. On the other hand, it was shown in [BeBoRö 06] that for any semi-Dirichlet form $(\mathcal{E},\mathcal{F})$ on a Lusin measurable space $E$, one can always find a larger space $E_{1}$ s.t. $E_{1}\setminus E$ has measure zero and $(\mathcal{E},\mathcal{F})$ regarded on $E_{1}$ becomes quasi-regular.

Hereinafter, all of the forms are assumed to be quasi-regular, in particular every element $u\in\mathcal{F}$ admits a quasi-continuous version denoted by $\widetilde{u}$.

In the sequel we often appeal to the following well known decompositions for the elements of the domain $\mathcal{F}$:

Ortogonal decomposition via hitting distribution. For a nearly Borel set $A\subset E$ and a quasi-continuous function $u\in\mathcal{F}$ we define the $\alpha$-order hitting distribution $R_{\alpha}^{A^{c}}u(x):=\mathbb{E}^{x}[e^{-\alpha T_{A^{c}}}u(X_{T_{A^{c}}})]$, $\alpha>0$. Then $R_{\alpha}^{A^{c}}u\in\mathcal{F}$ is quasi-continuous, $u-R_{\alpha}^{A^{c}}u\in\mathcal{F}_{A}$, and $\mathcal{E}_{\alpha}(R_{\alpha}^{A^{c}}u,v)=0$ for all $v\in\mathcal{F}_{A}$. When $\mathcal{E}$ is a Dirichlet form, $\widehat{R}_{\alpha}^{A^{c}}u$ may be defined analogously, replacing $X$ with the dual process $\widehat{X}$. When $\mathcal{E}$ is merely a semi-Dirichlet form, the existence of the dual process is more delicate, and for simplicity we prefere to define $\widehat{R}_{\alpha}^{A^{c}}u$ as the unique element from $\mathcal{F}$ such that $u-\widehat{R}_{\alpha}^{A^{c}}u\in\mathcal{F}_{A}$ and $\mathcal{E}_{\alpha}(v,R_{\alpha}^{A^{c}}u)=0$ for all $v\in\mathcal{F}_{A}$; see e.g. [Os 13], Section 3.5.

Fukushima’s decomposition. (see [MaRö 92], Chapter VI, Theorem 2.5, or [FuOsTa 11]) For each $u\in\mathcal{F}$ there exist a martingale additive functional of finite energy $(M_{t})_{t\geq 0}$ (MAF) and a continuous additive functional $(N_{t})_{t\geq 0}$ of zero energy s.t. $\widetilde{u}(X)-\widetilde{u}(X_{0})=M+N$; we denote by $|N|_{t}$ the variation of $N$ on $[0,t]$.

For the rest of this section our aim is to explore conditions for an element $u\in\mathcal{F}$ (or more generally in $\mathcal{F}_{loc}$) ensuring that $\widetilde{u}(X)$ is a $\mathbb{P}^{x}$-semi($\alpha$-quasi)martingale q.e. $x\in E$; in this case we shall say shortly that $\widetilde{u}(X)$* is a semi($\alpha$-quasi)martingale*.

Going back to Proposition 4.4 and Corollary 4.5, we note that the sub-Markov property of the dual semigroup was quite helpful and for this reason we shall first deal with Dirichlet forms. However, in the end of this section we shall see that the results can be extended to semi-Dirichlet forms in their full generality. It is worth to mention that all of the forthcoming criteria for quasimartingale functions can be directly transferred to lower bounded (semi) Dirichlet forms whose semigroups are associated to right processes, but for simplicity we deal only with (semi) Dirichlet forms.

Theorem 5.1**.**

The following assertions are equivalent for an element $u\in\mathcal{F}$.

i) $|\mathcal{E}(u,v)|\leq const\cdot(\|v\|_{\infty}+\|v\|_{2})$ for all $v\in\mathcal{F}_{b}$.

ii) For one (hence all) $\alpha>0$ there exist $u_{1},u_{2}\in E(\mathcal{U}^{\alpha})$ such that

ii.1) $\sup\limits_{\|f\|_{\infty}+\|f\|_{2}\leq 1}\int_{E}(u_{1}+u_{2})fd\mu<\infty$.

ii.2) $u=u_{1}-u_{2}$ $\mu$-a.e.

ii.3) $\sup\limits_{\|f\|_{\infty}+\|f\|_{2}\leq 1}\int_{E}|P_{t}u_{i}-u_{i}|fd\mu\leq{const}\cdot t$, $i=1,2$.

iii) For one (hence all) $\alpha>0$, $\widetilde{u}(X)$ is an $\alpha$-quasimartingale and

[TABLE]

for sufficiently small $t\geq 0$.

In particular, if $u$ satisfies i) then there exists a smooth measure $\nu$ such that $\mathcal{E}(u,v)=\nu(\widetilde{v})$ for all $v\in\mathcal{F}_{b}$.

Proof.

i) $\Rightarrow$ ii). Let $t>0$, $f\in L^{\infty}(\mu)\cap L^{2}(\mu)$, $w:={\rm sgn}(P_{t}u-u)f$, and set $v:=\int_{0}^{t}\widehat{P}_{s}wds$. Then

[TABLE]

By Proposition 4.4 (take $\mathcal{A}:=\{f\in L^{2}(\mu):\;\|f\|_{\infty}+\|f\|_{2}\leq 1\}$) we obtain ii).

ii) $\Rightarrow$ iii). Since $\widetilde{u}$ is quasi-continuous, we have $\widetilde{u}=u_{1}-u_{2}$ q.e., hence the first assertion is clear (by Corollary 3.2 for example).

Since $(e^{-\alpha t}u_{i}(X_{t}))_{t\geq 0}$, $i=\overline{1,2}$ are right continuous supermartingales, by [Sha 88], Section VI, there exist (uniquely) local martingales $M^{i}$, $M^{i}_{0}=0$, and predictable right continuous increasing and non-negative processes $A^{i}$ such that $e^{-\alpha t}u_{i}(X_{t})-u_{i}(X_{0})=M^{i}_{t}-A^{i}_{t}$, $t\geq 0$.

If $(T_{n}^{i})_{n}$ are stopping times increasing a.s. to infinity such that the stopped processes $(M^{i}_{t\wedge T_{n}^{i}})_{t\geq 0}$ are uniformly integrable martingales, we get

[TABLE]

Therefore, $\mathbb{E}^{x}[A_{t}^{i}]\leq u_{i}(x)-e^{-\alpha t}P_{t}u_{i}(x)$ and if $f\in L^{\infty}(\mu)\cap L^{2}(\mu)$

[TABLE]

$\leq{const}\cdot t(\|f\|_{\infty}+\|f\|_{2})$.

Also, $e^{-\alpha t}\widetilde{u}(X_{t})-\widetilde{u}(X_{0})=\overline{M_{t}}+\overline{A_{t}}$, where $\overline{M}:=M^{1}-M^{2}$ is a local martingale and $\overline{A}:=A^{2}-A^{1}$ is a predictable right continuous process of bounded variation.

On the other hand, since $\widetilde{u}(X)$ is an $\alpha$-quasimartingale, it follows that $N$, the CAF from Fukushima decomposition, is a continuous semimartingale, hence it is the sum of a local martingale and a continuous process with bounded variation (see e.g. [Pr 05], page 131). But $N$ has zero energy so the quadratic variation of its martingale part is zero, hence $N$ is of bounded variation. Then, integrating by parts,

[TABLE]

By the uniqueness of the canonical decomposition of $(e^{-\alpha t}\widetilde{u}(X_{t}))_{t\geq 0}$ we get that

[TABLE]

Therefore,

[TABLE]

and

[TABLE]

But by the previously obtained estimates for $\mathbb{E}^{f\cdot\mu}[A_{t}^{i}]$, we get

[TABLE]

$\leq{const}\cdot t(\|f\|_{\infty}+\|f\|_{2})$

for conveniently small $t$, since $\mathbb{E}^{f\cdot\mu}[|M_{t}|]\leq\|f\|_{2}\mathbb{E}^{\mu}[M_{t}^{2}]$ and $M$ is of finite energy (i.e. $\lim_{t\to 0}\frac{1}{t}\mathbb{E}^{\mu}[M_{t}^{2}]<\infty$).

iii) $\Rightarrow$ i). By Revuz correspondence (see [MaRö 92], Theorem 2.4), if $v=\widehat{U}_{\alpha}f$ for some $\alpha>0$ and $f\in L^{2}(\mu)\cap L^{\infty}(\mu)$

[TABLE]

where $\nu$ is the signed Revuz measure associated to $N$. By an approximation argument, $|\mathcal{E}(u,v)|=|\nu(\widetilde{v})|\leq{const}\;(\|v\|_{\infty}+\|v\|_{2})$ for all $v\in\mathcal{F}_{b}$. ∎

Further versions of Theorem 5.1 can be taken into account. For example, the following result extends Theorem 6.2 from [Fu 99] to the non-symmetric case and it can be proved in the same manner as Theorem 5.1, so we omit it.

Theorem 5.2**.**

The following assertions are equivalent for $u\in\mathcal{F}$.

i) $|\mathcal{E}(u,v)|\leq{const}\cdot\|v\|_{\infty}$ for all $v\in\mathcal{F}_{b}$.

ii) For each $\alpha>0$, $\widetilde{u}(X)$ is an $\alpha$-quasimartingale and $\mathbb{E}^{\mu}[|N|_{t}]\leq{const}\cdot t$ for small $t$.

iii) There exists a smooth signed measure (the Revuz measure of $N$) $\nu$ such that $\nu$ is finite and $\mathcal{E}(u,v)=\nu(\widetilde{v})$ for all $v\in\mathcal{F}$.

Now, we turn our attention to the situation when any of the equivalent assertions of Theorem 5.2 holds only locally. The following result extends Theorem 6.1 from [Fu 99] to the non-symmetric case.

Theorem 5.3**.**

The following assertions are equivalent for $u\in\mathcal{F}$.

i) $\widetilde{u}(X)$ is a semimartingale.

ii) There exists a nest $(F_{n})_{n\geq 1}$ and constants $c_{n}$ such that

[TABLE]

Proof.

i) $\Rightarrow$ ii). As in the poof of ii) $\Rightarrow$ iii) in Theorem 5.1, if $\widetilde{u}(X)$ is a semimartingale then $N$ (the CAF in Fukushima decomposition) is a continuous semimartingale of zero energy, hence it is of bounded variation. By [MaRö 92], Theorem 2.4, $N$ is in Revuz correspondence with a signed smooth measure $\nu$, with an attached nest of compacts $(F_{n})_{n\geq 1}$ s.t. $\nu(F_{n})<\infty$. Then just as in the proof of Theorem 5.4.2. in [FuOsTa 11], one obtains that $\mathcal{E}(u,v)=\nu(\widetilde{v})$ for all $v\in\mathcal{F}$.

ii) $\Rightarrow$ i). Without loss we can assume that $\mu(F_{n})<\infty$. Also, since $(F_{n})_{n}$ is a nest we have that $\mathop{\lim}\limits_{n}T_{F_{n}^{c}}\geq\xi$ a.s. Due to a result of Meyer (see e.g. [Pr 05], Theorem 6) it is sufficient to show that $(\widetilde{u}(X_{t})1_{[0,T_{F_{n}^{c}})}(t))_{t\geq 0}$ is a semimartingale for each $n$ (such an argument was also employed in [ÇiJaPrSh 80], after Theorem 4.6 ). On the other hand, $(e^{-t}R_{1}^{F_{n}^{c}}\widetilde{u}(X_{t})1_{[0,T_{F_{n}^{c}})}(t))_{t\geq 0}$ is a difference of two right continuous supermartingales, so we only have to check that $(\widetilde{u}-R_{1}^{F_{n}^{c}}\widetilde{u})(X)$ is a semimartingale. But, if $v\in\mathcal{F}_{b}$,

[TABLE]

$\leq(c_{n}+\int_{F_{n}}|u|d\mu)\|v-\widehat{R}_{1}^{F_{n}^{c}}v\|_{\infty}$

$\leq 2(c_{n}+\int_{F_{n}}|u|d\mu)\|v\|_{\infty}$,

and by Theorem 5.1 it follows that $(\widetilde{u}-R_{1}^{F_{n}^{c}}\widetilde{u})(X)$ is a semimartingale. ∎

Recall that $(\mathcal{E},\mathcal{F})$ is said to be local if for all pairs of elements $u,v\in\mathcal{F}$ with disjoint compact supports, it holds that $\mathcal{E}(u,v)=0$. By [MaRö 92], Chapter V, Theorem 1.5, $(\mathcal{E},\mathcal{F})$ is local if and only if the associated process is a diffusion.

When $\mathcal{E}$ is local, Theorem 5.3 remains true if $u$ is assumed to be only locally in $\mathcal{F}$. Actually, the following even more general statement holds.

Corollary 5.4**.**

Assume that $(\mathcal{E},\mathcal{F})$ is local. Let $u$ be a real-valued $\mathcal{B}$-measurable finely continuous function and let $(v_{k})_{k}\subset\mathcal{F}$ such that $v_{k}\mathop{\longrightarrow}\limits_{k\to\infty}u$ pointwise except a $\mu$-polar set and boundedly on each $F_{n}$. Further, suppose that there exist constants $c_{n}$ such that

[TABLE]

Then $u(X)$ is a semimartingale.

Proof.

As we already mentioned in the proof of Theorem 5.3, ii) $\Rightarrow$ i), it is sufficient to show that $(u-R_{1}^{F_{n}^{c}}u)(X)$ is a semimartingale. Also, we saw that $|\mathcal{E}_{1}(\widetilde{v}_{k}-R_{1}^{F_{n}^{c}}\widetilde{v}_{k},v)|\leq const\cdot\|v\|_{\infty}$ for all $v\in\mathcal{F}_{b}$, where the constant in the right-hand side may depend on $n$. Now, by Theorem 5.1 we have that by setting $\widetilde{v}_{k}^{n}:=\widetilde{v}_{k}-R_{1}^{F_{n}^{c}}\widetilde{v}_{k}$,

[TABLE]

But by hypothesis, $R_{1}^{F_{n}^{c}}\widetilde{v}_{k}(x)=\mathbb{E}^{x}[e^{-T_{F_{n}^{c}}}\widetilde{v}_{k}(X_{T_{F_{n}^{c}}})]$ converges $\mu$-a.s. and boundedly to $R_{1}^{F_{n}^{c}}u$ as $k$ tends to infinity. So by setting $u_{n}:=u-R_{1}^{F_{n}^{c}}u$ and by dominated convergence

[TABLE]

$\leq{const}\cdot t(\|f\|_{\infty}+\|f\|_{2})$

for all $f\in L^{1}(\mu)$. Therefore, by Proposition 4.4 we get that $u_{n}(X)$ is a semimartingale. ∎

5.1 Extensions to semi-Dirichlet forms

We reiterate that for the previous results of this section, where we considered only Dirichlet forms, it was used the fact that the adjoint semigroup $(\widehat{P}_{t})_{t\geq 0}$ was sub-Markovian; e.g. in order to have the estimate $\|\int_{0}^{t}\widehat{P}_{s}f\;ds\|_{\infty}\leq t\|f\|_{\infty}$. In this subsection we show that, as a matter of fact, the sub-Markov property of the adjoint semigroup is not crucial and most of the previous results remain valid for semi-Dirichlet forms. More precisely, although in order to extend theorems 5.1 and 5.2, i) $\Rightarrow$ ii) to semi-Dirichlet forms, essentially with the same proofs, it is sufficient to assume the existence of a strictly positive bounded co-excessive function, Theorem 5.3, $ii)\Rightarrow i)$ remains true without any further assumptions, due to a standard localization procedure. Finally, before we present the announced extensions, we emphasize once again that the case of lower bounded semi-Dirichlet forms follows easily, by working with $\mathcal{E}_{\alpha}$ instead of $\mathcal{E}$, for instance.

Hereinafter, we keep the same context and notations as before, but we assume that $(\mathcal{E},\mathcal{F})$ is merely a (quasi-regular) semi-Dirichlet form on $L^{2}(E,\mu)$, i.e. we drop condition iv) from the beginning of this section.

Before we present the announced extension, in order to fix the notations, let us recall the following localization procedure: Let $G$ be a finely open set and consider the bilinear form

[TABLE]

Then by [BeBo 04], Theorem 7.6.11 (see also [Os 13], Theorem 3.5.7), $(\mathcal{E}^{G},\mathcal{F}_{G})$ is a (quasi-regular) semi-Dirichlet form whose associated process is $X^{G}$ with state space $G\cup\{\Delta\}$, obtained by killing $X$ upon leaving $G$:

[TABLE]

The associated semigroup and resolvent are denoted by $(P_{t}^{G})_{t\geq 0}$ and $(U_{\alpha}^{G})_{\alpha>0}$.

Theorem 5.5**.**

Let $u\in\mathcal{F}$ and assume there exist a nest $(F_{n})_{n\geq 1}$ and constants $(c_{n})_{n\geq 1}$ such that

[TABLE]

Then $\widetilde{u}(X)$ is a semimartingale.

Proof.

Let us fix a quasi-continuous element $0<f_{0}\in\mathcal{F}$ and a sequence of positive constants $\alpha_{k}\nearrow_{k}\infty$. By [MaRö 92], Theorem 2.13 we have that $\alpha_{k}\widehat{U}_{\alpha_{k}}f_{0}\mathop{\longrightarrow}\limits_{k}^{\mathcal{E}_{1}^{1/2}}f_{0}$, hence by [MaOvRö 95], Proposition 2.18, (i), there exists a nest $(F^{\prime}_{n})_{n\geq 1}$ s.t. (by passing to a subsequence if necessary) $\lim\limits_{k}\alpha_{k}\widehat{U}_{\alpha_{k}}f_{0}=f_{0}$ uniformly on each $F^{\prime}_{n}$. Consequently, replacing $F_{n}$ with $F_{n}\cap F^{\prime}_{n}$, we may assume that $(F_{n})_{n\geq 1}$ is a nest such that $\sup\limits_{k}\|1_{F_{n}}\alpha_{k}\widehat{U}_{\alpha_{k}}f_{0}\|_{\infty}<\infty$. Also, without loss of generality we suppose that $\mu(F_{n})<\infty$ for all $n\geq 1$.

Now, let us consider the fine interiors $G_{n}:=\accentset{\circ}{F}_{n}^{f}$ and the localized semi-Dirichlet forms $(\mathcal{E}^{G_{n}},\mathcal{F}_{G_{n}})$. As before, the idea is to localize $u$ by setting

[TABLE]

so that by setting $c_{k}^{n}:=c_{n}+\alpha_{k}\|u_{n}\|_{L^{1}(G_{n},\mu)}+\|u\|_{L^{1}(G_{n},\mu)}$, for all $v\in\mathcal{F}_{G_{n}}$

[TABLE]

On the one hand, we claim that $(u_{n}(X_{t})1_{[0,T_{G_{n}^{c}})}(t))_{t\geq 0}$ is a $\mathbb{P}^{x}$-semimartingale q.e. $x\in G_{n}$. To see this, let us introduce for all $\alpha>0$ and $n\geq 1$

[TABLE]

and note that $v_{k}^{n}$ is $\mathcal{E}^{G_{n}}_{\alpha_{k}}$-co-excessive (i.e. $\widehat{P}_{s}^{G_{n},\alpha_{k}}v_{k}^{n}\leq v_{k}^{n}$) and $v_{k}^{n}\mathop{\longrightarrow}\limits_{k}^{\mathcal{E}^{G_{n}}_{1}}f_{0}|_{G_{n}}>0$. Furthermore, by the way we chose the nest $(F_{n})_{n\geq 1}$

[TABLE]

It follows that for all $r,t>0$,

$\int_{G_{n}}P_{r}^{G_{n},\alpha_{k}+1}|P_{t}^{G_{n},\alpha_{k}+1}u_{n}-u_{n}|v_{k}^{n}\;d\mu=|\mathcal{E}_{\alpha_{k}+1}^{G_{n}}(u_{n},\int_{0}^{t}\widehat{P}_{s}^{G_{n},\alpha_{k}+1}(sgn(P_{t}^{G_{n},\alpha_{k}+1}u_{n}-u_{n})\widehat{P}_{r}^{G_{n},\alpha_{k}+1}v_{k}^{n}\;ds))|$

$\leq c_{k}^{n}\|\int_{0}^{t}\widehat{P}_{s}^{G_{n},{\alpha_{k}+1}}(sgn(P_{t}^{G_{n},{\alpha_{k}+1}}u_{n}-u_{n})\widehat{P}_{r}^{G_{n},{\alpha_{k}+1}}v_{k}^{n}\;ds)\|_{\infty}$

$\leq c_{k}^{n}\int_{0}^{t}\|\widehat{P}_{s+r}^{G_{n},{\alpha_{k}+1}}v_{k}^{n}\|_{\infty}\;ds\leq c_{k}^{n}\int_{0}^{t}e^{-(r+s)}\|v_{k}^{n}\|_{\infty}\;ds$

$\leq c_{k}^{n}d_{n}e^{-r}t$.

Let now $\tau_{l}:=\{\frac{i}{2^{l}}:0\leq i\leq l2^{l}\}$, $l\geq 1$. As in the proof of Proposition 4.4, i) $\Rightarrow$ ii), we get

[TABLE]

hence, by e.g. Proposition 4.2, the process $(e^{-(\alpha_{k}+1)t}u_{n}(X^{G_{n}}_{t}))_{t\geq 0}$, and more importantly, the process $u_{n}(X^{G_{n}})$, are $\mathbb{P}^{x}$-semimartingales for $v_{k}^{n}\cdot\mu$-a.e. $x\in G_{n}$ for all $k>0$. This means that $(u_{n}(X_{t})1_{[0,T_{G_{n}^{c}})}(t))_{t\geq 0}$ is a $\mathbb{P}^{x}$-semimartingale q.e. $x\in G_{n}$.

On the other hand, as in the proof of Theorem 5.3, ii) $\Rightarrow$ i), the process $(R_{1}^{G_{n}^{c}}\widetilde{u}(X_{t})1_{[0,T_{G_{n}^{c}})})_{t\geq 0}$ is a semimartingale, hence $\widetilde{u}(X_{t})1_{[0,T_{G_{n}^{c}})}(t)=u_{n}(X_{t})1_{[0,T_{G_{n}^{c}})}(t)+R_{1}^{G_{n}^{c}}\widetilde{u}(X_{t})1_{[0,T_{G_{n}^{c}})}$, $t\geq 0$ is a semimartingale. By the result of Meyer already used in Theorem 5.3, it is sufficient to show that $\mathop{\lim}\limits_{n}T_{G_{n}^{c}}\geq\xi$ a.s. But this property is true for $(F_{n})_{n\geq 1}$ and it is inherited by $(G_{n})_{n\geq 1}$ because for any $f\in E(\mathcal{U}^{1})$, $R_{1}^{F_{n}^{c}}f=R_{1}^{G_{n}^{c}}f$.

∎

The case when $u$ is merely locally in $\mathcal{F}$ can be treated just like Corollary 5.4, so we state this observation as a corollary, but we omit the proof.

Corollary 5.6**.**

The statement of Corollary 5.4 remains valid if $(\mathcal{E},\mathcal{F})$ is a semi-Dirichlet form.

5.2 Final remarks

For practical reasons, it is useful to know whether it is sufficient to check inequalities (1.2) and (1.1) only for $v\in\mathcal{F}_{0}$, where $\mathcal{F}_{0}$ is a certain proper subspace of $\mathcal{F}$. We point out below some ideas of choosing $\mathcal{F}_{0}$.

a) Assume that $(\mathcal{E},\mathcal{F})$ is a (non-symmetric) Dirichlet form and take $\mathcal{F}_{0}:=D({\sf\widehat{L}})\cap L^{\infty}(\mu)$. If inequality (1.2) is verified for all $v\in\mathcal{F}_{0}$ then $\widetilde{u}(X)$ is an $\alpha$-quasimartingale for all $\alpha>0$. This is true by Proposition 4.5.

b) Assume that $(\mathcal{E},\mathcal{F})$ is a semi-Dirichlet form. We consider the following extension of condition $(\mathcal{L})$ from [Fu 99]: a subspace $\mathcal{F}_{0}\subset\mathcal{F}_{b}$ satisfies condition $(\mathcal{S})$ if $\mathcal{F}_{0}$ is $\mathcal{E}_{1}^{1/2}$-dense in $\mathcal{F}$ and there exists a bounded continuous function $\phi:\mathbb{R}\mapsto\mathbb{R}$ such that

• $\phi(\mathcal{F}_{0})\subset\mathcal{F}_{0}$;

• $\phi(t)=t$ if $t\in[-1,1]$;

• if $(v_{n})_{n\geq 1}\subset\mathcal{F}_{0}$ is $\mathcal{E}_{1}^{1/2}$-convergent then $(\phi(v_{n}))_{n}$ is $\mathcal{E}_{1}^{1/2}$-bounded.

As a candidate for a space satisfying condition ($\mathcal{S}$), one should have in mind a core in the sense of [FuOsTa 11] (for regular Dirichlet forms), or the space of cylindrical functions in the infinite dimensional situation (see e.g. [MaRö 92]), while $\phi$ could be a smooth unit contraction.

In the same spirit as [Fu 99], Lemma 6.1, we have the following result.

Lemma 5.7**.**

Let $u\in\mathcal{F}$ and $\mathcal{F}_{0}$ satisfy condition ($\mathcal{S}$). If inequality (1.2) holds for all $v\in\mathcal{F}_{0}$ then it holds for all $v\in\mathcal{F}_{b}$.

Proof.

Let $v\in\mathcal{F}$ and assume that $\|v\|_{\infty}\leq 1$. If $(v_{n})_{n\geq 1}\subset\mathcal{F}_{0}$ is $\mathcal{E}_{1}^{1/2}$-convergent to $v$, then from the boundedness condition on $(\phi(v_{n}))_{n}$ and by Banach-Sacks theorem, there exists a subsequence $(\phi(v_{n_{k}}))_{n_{k}}$ whose Cesaro means $\frac{i=1}{k}\sum\limits_{1}^{k}\phi(v_{n_{i}})$ converges to $\phi(v)=v$ w.r.t. $\mathcal{E}_{1}^{1/2}$. Therefore $|\mathcal{E}(u,v)|=\lim\limits_{k}|\mathcal{E}(u,\frac{1}{k}\sum\limits_{i=1}^{k}\phi(v_{n_{i}}))|\leq c\|\phi\|_{\infty}$. ∎

Regarding inequality (1.1), we have:

Lemma 5.8**.**

Let $u\in\mathcal{F}$ and $\mathcal{F}_{0}$ satisfy condition ($\mathcal{S}$) such that inequality (1.1) holds for $\mathcal{F}_{b,F_{n}}$ replaced by $\mathcal{F}_{b,F_{n}}\cap\mathcal{F}_{0}$. In addition, assume that $\mathcal{F}_{0}$ is an algebra and that for each $n\geq 1$ there exists $\psi_{n}\in\mathcal{F}_{0}\cap\mathop{\bigcup}\limits_{k}\mathcal{F}_{b,F_{k}}$ such that $\psi_{n}=1$ on $F_{n}$. Then inequality (1.1) holds for all $n\geq 1$ and $v\in\mathcal{F}_{b,F_{n}}$, with possible different constants $c_{n}$.

Proof.

Fix $n\geq 1$, $v\in\mathcal{F}_{F_{n}}$ s.t. $\|v\|_{\infty}\leq 1$, and let $k(n)\geq 1$ s.t. $\psi_{n}\in\mathcal{F}_{b,F_{k(n)}}$. Take $(v_{m})_{m}\subset\mathcal{F}_{0}$ which is $\mathcal{E}_{1}^{1/2}$-convergent to $v$. Then

[TABLE]

which means that $(\phi(v_{m})\psi_{n})_{m}$ is $\mathcal{E}_{1}^{1/2}$-bounded and employing once again Banach-Sacks theorem just like we did in the proof of the previous lemma, we get

[TABLE]

where the right-hand term does not depend on $v$ (in fact it is the new constant replacing $c_{n}$).

∎

Candidates for $\mathcal{F}_{0}$ satisfying the assumption of Lemma 5.8 are the special standard cores in the sense of [FuOsTa 11]; see also [Fu 99], page 27.

Aknowledgements. The first named author acknowledges support from the Romanian National Authority for Scientific Research, project number PN-III-P4-ID-PCE-2016-0372. The second-named author acknowledges support from the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-RU-TE-2014-4-0007.