Hunt's Hypothesis (H) for Markov Processes: Survey and Beyond

TL;DR

This paper surveys Hunt's hypothesis (H) for Markov processes and Le9vy processes, explores multidimensional cases, and discusses open questions for future research.

Contribution

It provides a comprehensive survey of Hunt's hypothesis (H), investigates multidimensional Le9vy processes, and highlights open problems in the field.

Findings

Survey of existing results on Hunt's hypothesis (H)

Analysis of (H) for multidimensional Le9vy processes

Identification of open questions for future research

Abstract

The goal of this paper is threefold. First, we survey the existing results on Hunt's hypothesis (H) for Markov processes and Getoor's conjecture for L\'{e}vy processes. Second, we investigate (H) for multidimensional L\'{e}vy processes from the viewpoints of projections and energy, respectively. Third, we present a few open questions for further study.

Equations

$$T_D:=\inf \{t>0:X_t\in D\}.$$

$$(P_{\alpha }^{\ast })\iff (E_{\alpha }^{\ast })\iff (M_{\alpha }^{\ast })\iff (R_{\alpha }^{\ast })\iff (\mathrm{H}).$$

$$E[\exp \{i⟨z,X_t⟩\}]=\exp \{-t\psi (z)\},z\in {\mathbf{R}}^n,t\ge 0.$$

$$\psi (z)=i⟨a,z⟩+\frac{1}{2}⟨z,Qz⟩+{\int }_{{\mathbf{R}}^n}\left(1-e^{i⟨z,x⟩}+i⟨z,x⟩1_{\{|x|<1\}}\right)\mu (dx),$$

$$A:=1+\mathrm{Re}(\psi ),B:=|1+\psi |.$$

$$\hat{\nu }(z):={\int }_{{\mathbf{R}}^n}{e}^{i⟨z,x⟩}\nu (dx).$$

$${\int }_{{\mathbf{R}}^n}\frac{A(z)}{B^2(z)}|\hat{\nu }(z){|}^{2}dz<\infty .$$

$${\int }_0^{\infty }\text{mbox}Re([1+\psi (z){]}^{-1})dz<\infty .$$

$$\underset{\lambda \to \infty }{\lim }{\int }_{{\mathbf{R}}^n}\frac{\lambda +\mathrm{Re}\psi (z)}{|\lambda +\psi (z){|}^2}|\hat{\nu }(z){|}^{2}dz$$

$$\mathrm{Re}\psi (z)\ge c⟨z,z⟩,z\in {\mathbf{R}}^n,$$

$${\int }_{{\mathbf{R}}^n}\frac{|{\psi }_2(z)|}{(1+\mathrm{Re}\psi (z){)}^2+(\mathrm{Im}\psi (z){)}^2}dz<\infty ,$$

$$\underset{|z|\to \infty }{\mathrm{liminf}}\frac{|\psi (z)|}{|z|{\log }^{1+\gamma }|z|}>0$$

$$\underset{\lambda \to \infty }{\lim }\sum _{k=1}^{\infty }{\int }_{\left\{B(z)>A(z)f(A(z)),k\le \frac{|\mathrm{Im}\psi (z)|}{A(z)}<k+1,A(z)\le \lambda <(k+1)|\mathrm{Im}\psi (z)|\right\}}\frac{\lambda }{{\lambda }^2+(\mathrm{Im}\psi (z){)}^2}|\hat{\nu }(z){|}^{2}dz=0$$

$$Condition(C^{\log }):Foranyfinitemeasure\nu on{\mathbf{R}}^{n}offinite1\text{-}energy,there$$

$$existaconstant\varsigma >1andasequence\{y_k\uparrow \infty \}suchthat{y}_1>1and$$

$$\sum _{k=1}^{\infty }{\int }_{\{y_k\le B(z)<(y_k{)}^{\varsigma }\}}\frac{1}{B(z)\log (B(z))}|\hat{\nu }(z){|}^{2}dz<\infty .$$

$$Condition(C^{\log \log }):Foranyfinitemeasure\nu on{\mathbf{R}}^{n}offinite1\text{-}energy,$$

$$thereexistaconstant\varsigma >1andasequenceofpositivenumbers\{x_k\}$$

$$suchthat{N}_{x_1}^{\varsigma }>e,x_k+1<x_{k+1},k\in \mathbf{N},\sum _{k=1}^{\infty }\frac{1}{x_k}=\infty ,and$$

$$\sum _{k=1}^{\infty }{\int }_{\{N_{x_k}^{\varsigma }\le B(z)<N_{x_k+1}^{\varsigma }\}}\frac{1}{B(z)\log (B(z))[\log \log (B(z))]}|\hat{\nu }(z){|}^{2}dz<\infty .$$

$$\underset{\lambda \to \infty }{\lim }{\int }_{{\mathbf{R}}^n}\frac{\lambda }{{\lambda }^2+B^2(z)}|\hat{\nu }(z){|}^{2}dz=0$$

$$\underset{\epsilon \to 0}{\mathrm{liminf}}\frac{{\int }_{-\epsilon }^{\epsilon }{x}^2\mu (dx)}{\epsilon /|\log \epsilon |}>0,$$

$$\underset{\lambda \to \infty }{\lim }{\int }_{\{B(z)>A(z)f(A(z))\}}\frac{\lambda }{{\lambda }^2+B^2(z)}|\hat{\nu }(z){|}^{2}dz=0$$

$$\xi (dx):=(1\wedge |x|)\mu (dx),x\in \mathbf{R}.$$

$$\underset{|z|\to \infty }{\mathrm{liminf}}\frac{\mathrm{Re}\psi (z)}{|z|/\log |z|}>0,$$

$$\underset{\epsilon \to 0}{\mathrm{liminf}}\frac{{\int }_{-\epsilon }^{\epsilon }{x}^2\mu (dx)}{\frac{\epsilon }{|\log \epsilon |[\log |\log \epsilon |]}}>0,$$

$$\psi (z)=-idz+{\int }_0^{\infty }\left(1-e^{izx}\right)\mu (dx),z\in \mathbf{R},$$

$$U(A)=E\left[{\int }_0^{\infty }{1}_A(X_t)dt\right],A\subset [0,\infty ).$$

$$c_1{\mu }_S-{\mu }_1\le \mu \le c_2{\mu }_S+{\mu }_2\mathrm{on}(0,\delta ).$$

$$\frac{1}{c{x}^{1+\alpha }}\le \rho (x)\le \frac{c}{x^{1+\beta }},\forall x\in (0,1].$$

$$\zeta (\omega )=\inf \{\omega :Z_t(\omega )=\Delta \},\omega \in \Omega .$$

$$(H_m):\text{mbox}everysemipolarsetism\text{mbox}-essentiallypolar.$$

$${\int }_{(\sqrt{Q}{\mathbf{R}}^n{)}^{\perp }}(1\wedge |x|){\mu }_{P_2}(dx)<\infty .$$

Full text

Ze-Chun Hu

College of Mathematics

Sichuan University

Chengdu, 610065, China

E-mail: zchu@scu.edu.cn

Wei Sun

Department of Mathematics and Statistics

Concordia University

Montreal, H3G 1M8, Canada

E-mail: wei.sun@concordia.ca

1 Introduction

Let $E$ be a locally compact space with a countable base and $X=(X_{t},P^{x})$ be a standard Markov process on $E$ as described in Blumenthal and Getoor [2]. Denote by ${\mathcal{B}}$ and ${\mathcal{B}}^{n}$ the family of all Borel measurable subsets and nearly Borel measurable subsets of $E$, respectively. For $D\subset E$, we define the first hitting time of $D$ by

[TABLE]

A set $D\subset E$ is called thin if there exists a set $C\in{\mathcal{B}}^{n}$ such that $D\subset C$ and $P^{x}(T_{C}=0)=0$ for any $x\in E$. $D$ is called semipolar if $D\subset\bigcup_{n=1}^{\infty}D_{n}$ for some thin sets $\{D_{n}\}_{n=1}^{\infty}$. $D$ is called polar if there exists a set $C\in{\mathcal{B}}^{n}$ such that $D\subset C$ and $P^{x}(T_{C}<\infty)=0$ for any $x\in E$. Let $m$ be a measure on $(E,{\mathcal{B}})$. $D$ is called $m$-essentially polar if there exists a set $C\in{\mathcal{B}}^{n}$ such that $D\subset C$ and $P^{m}(T_{C}<\infty)=0$. Hereafter $P^{m}(\cdot):=\int_{E}P^{x}(\cdot)m(dx)$.

Hunt’s hypothesis (H) says that “every semipolar set of $X$ is polar”. This hypothesis plays a crucial role in the potential theory of (dual) Markov processes. It is known that if $X$ is in duality with another standard process $\hat{X}$ on $E$ with respect to a $\sigma$-finite reference measure $m$, then (H) is equivalent to many potential principles for Markov processes. Denote by $E^{x}$ the expectation with respect to $P^{x}$. Let $\alpha>0$. A finite $\alpha$-excessive function $f$ on $E$ is called a regular potential provided that $E^{x}\{e^{-\alpha T_{n}}f(X_{T_{n}})\}\rightarrow E^{x}\{e^{-\alpha T}f(X_{T})\}$ for any $x\in E$ whenever $\{T_{n}\}$ is an increasing sequence of stopping times with limit $T$. Denote by $(U^{\alpha})_{\alpha>0}$ the resolvent operators for $X$.

• •

Bounded positivity principle $(P^{*}_{\alpha})$: If $\nu$ is a finite signed measure such that $U^{\alpha}\nu$ is bounded, then $\nu U^{\alpha}\nu\geq 0$, where $\nu U^{\alpha}\nu:=\int_{E}U^{\alpha}\nu(x)\nu(dx)$.

• •

Bounded energy principle $(E^{*}_{\alpha})$: If $\nu$ is a finite measure with compact support such that $U^{\alpha}\nu$ is bounded, then $\nu$ does not charge semipolar sets.

• •

Bounded maximum principle $(M^{*}_{\alpha})$: If $\nu$ is a finite measure with compact support $K$ such that $U^{\alpha}\nu$ is bounded, then $\sup\{U^{\alpha}\nu(x):x\in E\}=\sup\{U^{\alpha}\nu(x):x\in K\}$.

• •

Bounded regularity principle $(R^{*}_{\alpha})$: If $\nu$ is a finite measure with compact support such that $U^{\alpha}\nu$ is bounded, then $U^{\alpha}\nu$ is regular.

Theorem 1.1

(Blumenthal and Getoor [2, 3], Rao [31] and Fitzsimmons [6]) Assume that all 1-excessive (equivalently, all $\alpha$-excessive, $\alpha>0$) functions are lower semicontinuous. Then

[TABLE]

Hunt’s hypothesis (H) is also equivalent to some other important properties of Markov processes. For example, (H) holds if and only if the fine and cofine topologies differ by polar sets (Blumenthal and Getoor [3, Proposition 4.1] and Glover [13, Theorem 2.2]); (H) holds if and only if every natural additive functional of $X$ is a continuous additive functional (Blumenthal and Getoor [2, Chapter VI]); (H) is equivalent to the dichotomy of capacity (Fitzsimmons and Kanda [9]), which means that each compact set $K$ contains two disjoint sets with the same capacity as $K$.

In spite of its importance, (H) has been verified only in some special situations. Some fifty years ago, Getoor conjectured that essentially all Lévy processes satisfy (H), except for some extremely nonsymmetric cases like uniform motions. This conjecture stills remains open and is a major unsolved problem in the potential theory for Markov processes.

The rest of this paper is organized as follows. In Section 2, we survey the existing results on Hunt’s hypothesis (H) for Markov processes and Getoor’s conjecture for Lévy processes. In Sections 3 and 4, we investigate (H) for multidimensional Lévy processes from the viewpoints of projections and energy, respectively. In Section 5, we present a few open questions for further study.

2 Survey on (H) for Markov processes

In this section, we summarize the results that have been obtained so far for the validity of Hunt’s Hypothesis (H). We divide them into two parts: §2.1 (H) for Lévy processes and §2.2 (H) for Markov processes.

2.1 (H) for Lévy processes

Throughout this subsection, we let $(\Omega,{\mathcal{F}},P)$ be a probability space and $X=(X_{t})_{t\geq 0}$ be an $\mathbf{R}^{n}$-valued Lévy process on $(\Omega,{\mathcal{F}},P)$ with Lévy-Khintchine exponent $\psi$, i.e.,

[TABLE]

Hereafter we use $E$ to denote the expectation with respect to $P$, and use $\langle\cdot,\cdot\rangle$ and $|\cdot|$ to denote the Euclidean inner product and norm of $\mathbf{R}^{n}$, respectively. The classical Lévy-Khintchine formula tells us that

[TABLE]

where $a\in\mathbf{R}^{n},Q$ is a symmetric nonnegative definite $n\times n$ matrix, and $\mu$ is a measure (called the Lévy measure) on $\mathbf{R}^{n}\backslash\{0\}$ satisfying $\int_{\mathbf{R}^{n}\backslash\{0\}}(1\wedge|x|^{2})\mu(dx)<\infty$. For $x\in\mathbf{R}^{n}$, we denote by $P^{x}$ the law of $x+X$ under $P$. In particular, $P^{0}=P$. Denote by $m_{n}$ the Lebesgue measure on $\mathbf{R}^{n}$.

We use Re$(\psi)$ and Im$(\psi)$ to denote respectively the real and imaginary parts of $\psi$, and use also $(a,Q,\mu)$ to denote $\psi$. Define

[TABLE]

For a finite (positive) measure $\nu$ on $\mathbf{R}^{n}$, we denote

[TABLE]

$\nu$ is said to have finite 1-energy if

[TABLE]

We use $\log$ to denote $\log_{e}$.

2.1.1 Main results obtained before 1990

Suppose that $X$ is a compound Poisson process. Then every $x\in\mathbf{R}^{n}$ is regular for $\{x\}$, i.e., $P^{x}(T_{\{x\}}=0)=1$. Hence only the empty set is a semipolar set and therefore (H) holds.

When $n=1$, Kesten [29] (see also Bretagnolle [5]) showed that if $X$ is not a compound Poisson process, then every $\{x\}$ is non-polar if and only if

[TABLE]

It follows that if condition (2.1) is fulfilled, then (H) holds is equivalent to that only the empty set is a semipolar set.

Port and Stone [30] proved that for the asymmetric Cauchy process on the line every $x\in\mathbf{R}^{n}$ is regular for $\{x\}$. It follows that (H) holds in this case. Further, Blumenthal and Getoor [3] showed that all stable processes with index $\alpha\in(0,2)$ on the line satisfy (H). Kanda and Forst proved independently the following celebrated result.

Theorem 2.1

(Kanda [25] and Forst [10]) If $X$ has bounded continuous transition densities with respect to $m_{n}$ and $|\mbox{Im}(\psi)|\leq M(1+\mbox{Re}(\psi))$ for some positive constant $M$, then $X$ satisfies (H).

Rao [31] gave a short proof of the above Kanda-Forst theorem under the weaker condition that $X$ has resolvent densities with respect to $m_{n}$. In fact, Rao proved that the bounded maximum principle holds for $X$ in this case. By the Kanda-Forst theorem, we know that for $n\geq 1$ all stable processes with index $\alpha\neq 1$ satisfy (H). For the case $\alpha=1$, under the additional assumption that the linear term vanishes, Kanda [26] showed that (H) holds by virtue of the following result.

Theorem 2.2

(Kanda [26, Theorem 1]) Assume that $X$ has bounded continuous transition densities with respect to $m_{n}$. Then, a set contains a semipolar subset for $X$ which is not polar for $X$, if and only if it contains a nonpolar compact subset $H$ for $X$ such that the $\alpha$-capacity of $H$ for $X$ is uniformly bounded as $\alpha\uparrow\infty$.

By Hawkes [19, Theorem 2.1], we know that a Lévy process has resolvent densities with respect to $m_{n}$ if and only if all 1-excessive functions are lower semicontinuous. In [33], Rao gave a remarkable extension of the Kanda-Forst theorem.

Theorem 2.3

(Rao [33, Theorem 2])* Assume that $X$ has resolvent densities with respect to $m_{n}$. Suppose there is an increasing function $f$ on $[1,\infty)$ such that $\int_{N}^{\infty}(\lambda f(\lambda))^{-1}d\lambda=\infty$ for any $N\geq 1$ and $|1+\psi|\leq(1+{\rm Re}(\psi))f(1+{\rm Re}(\psi))$. Then (H) holds.*

The function $f$ in Theorem 2.3 can be taken as: (i) $f\equiv M$ for some positive constant $M$ (now Rao’s condition is reduced to the Kanda-Forst condition); (ii) $f(x)=\ln x$ for $x\geq 1$; (iii) $f(x)=\ln(1+\ln x)$ for $x\geq 1$. By Theorem 2.3, we find that the assumption “the linear term vanishes” put in [26] can be removed. Hence all stable processes on $\mathbf{R}^{n}$ satisfy (H).

To prove Theorem 2.3, Rao made use of the following result, which is a consequence of Rao [32, Lemma 2.1 and Theorem 1.5].

Theorem 2.4

(Rao [33, Theorem 1])* Assume that $X$ has resolvent densities with respect to $m_{n}$. Let $\nu$ be a finite measure of finite 1-energy. Then*

[TABLE]

exists. The limit is zero if and only if $U^{1}\mu$ is regular.

2.1.2 Main results obtained by our group

In this subsection, we introduce the main results obtained by our group on (H) for Lévy processes. We refer the reader to Hu and Sun [20], Hu et al. [24], Hu and Sun [21], Hu and Sun [22], and Hu et al. [23] for more details. The results are presented for three cases: (H) for general Lévy processes, (H) for one-dimensional Lévy processes, and (H) for subordinators.

(1) (H) for general Lévy processes

Theorem 2.5

*([20, Theorem 1.1]) Suppose that $Q$ is non-degenerate, i.e., $Q$ is of full rank. Then:

(i) $X$ satisfies (H);

(ii) The Kanda-Forst condition $|\mbox{Im}(\psi)|\leq M(1+\mbox{Re}(\psi))$ holds for some positive constant $M$;

(iii) $X$ and $\tilde{X}$ have the same polar sets, where $\tilde{X}=X-\bar{X}$ with $\bar{X}$ being an independent copy of $X$.*

We first proved (ii) by showing that

[TABLE]

for some positive constant $c$ and using the inequality that $|t-\sin t|\leq t^{2}/2$, $t\in\mathbf{R}$. By (2.2) and Hartman and Wintner [18], one finds that $X$ has bounded continuous transition densities. Hence (i) holds by the Kanda-Forst theorem. (iii) is a consequence of (ii), Kanda [25, Theorem 1] (or Hawkes [19, Theorems 2.1 and 3.3]), and the fact that $X$ has bounded continuous transition densities.

Denote $b:=-a$ and $\mu_{1}:=\mu|_{\mathbf{R}^{n}\backslash\sqrt{Q}\mathbf{R}^{n}}$. If $\int_{\{|x|<1\}}|x|\mu_{1}(dx)<\infty$, we set $b^{\prime}:=b-\int_{\{|x|<1\}}x\mu_{1}(dx)$. Define the following solution condition:

(S)  The equation $\sqrt{Q}y=b^{\prime},\ y\in\mathbf{R}^{n}$, has at least one solution.

Note that (S) means that $b^{\prime}\in\mathcal{R}(\sqrt{Q}):=\{\sqrt{Q}y:y\in\mathbf{R}^{n}\}$.

Theorem 2.6

([20, Theorem 1.2]) Suppose that $\mu({\mathbf{R}^{n}\backslash\sqrt{Q}\mathbf{R}^{n}})<\infty$. Then, the following three claims are equivalent:

(i) $X$ satisfies (H);

(ii) (S) holds;

(iii) The Kanda-Forst condition $|{\rm Im}(\psi)|\leq M(1+{\rm Re}(\psi))$ holds for some positive constant $M$.

The Lévy-Itô decomposition, orthogonal transformation and strong Markov property of Lévy processes, and some properties of compound Poisson processes have been used to prove Theorem 2.6.

Proposition 2.7

([20, Proposition 1.5]) Suppose that $X$ has bounded continuous transition densities, and $X$ and $\tilde{X}$ have the same polar sets, where $\tilde{X}$ is defined in Theorem 2.5(iii). Then $X$ satisfies (H).

To prove the above proposition, we used mainly an idea given in the proof of Kanda [26, Theorem 2] and the comparison inequality for capacities given in Kanda [25] (cf. also Hawkes [19]).

In [33], Rao mentioned that his condition $|1+\psi|\leq(1+{\rm Re}(\psi))f(1+{\rm Re}(\psi))$, equivalently, $|{\rm Im}(\psi)|\leq(1+{\rm Re}(\psi))f(1+{\rm Re}(\psi))$, is not far from being necessary for the validity of (H). The following result tells us that Rao’s condition can be relaxed.

Theorem 2.8

([24, Theorem 4.5]) Assume that all 1-excessive functions are lower semicontinuous. Then (H) holds if the following extended Kanda-Forst-Rao condition holds:

(EKFR) There are two measurable functions $\psi_{1}$ and $\psi_{2}$ on $\mathbf{R}^{n}$ such that $\rm{Im}(\psi)=\psi_{1}+\psi_{2}$, $|\psi_{1}|\leq Af(A)$, and

[TABLE]

where $f$ is a positive increasing function on $[1,\infty)$ such that $\int_{N}^{\infty}(\lambda f(\lambda))^{-1}d\lambda=\infty$ for some $N\geq 1$.

By virtue of Theorem 2.8, we constructed a class of one-dimensional Lévy processes satisfying (H) in [24, Example 4.8]. These Lévy processes have sufficient number of small jumps and no restriction is put on $a$ or $Q$. As another application of Theorem 2.8, we have the following result.

Proposition 2.9

([24, Proposition 4.10]) Let $X$ be a Lévy process on $\mathbf{R}$ such that all 1-excessive functions are lower semicontinuous. Suppose that

[TABLE]

for some constant $\gamma>0$. Then (H) holds.

To prove Theorem 2.8, we used the following necessary and sufficient condition for (H).

Theorem 2.10

([24, Theorem 4.3]) Assume that all 1-excessive functions are lower semicontinuous. Let $f$ be a positive increasing function on $[1,\infty)$ such that $\int_{N}^{\infty}(\lambda f(\lambda))^{-1}d\lambda=\infty$ for some $N\geq 1$. Then (H) holds if and only if

[TABLE]

for any finite measure $\nu$ with compact support such that $U^{1}\nu$ is bounded.

The proof of Theorem 2.10 is based on a key lemma ([24, Lemma 4.2]), which is obtained by using [33, Theorems 1 and 2].

The following two theorems provide new necessary and sufficient conditions for the validity of (H) for Lévy processes. Different from the classical Kanda-Forst condition and Rao’s condition, our conditions only require that ${\rm Im}(\psi)$ is partially well-controlled by $1+{\rm Re}(\psi)$. The weaker conditions are fulfilled by more general Lévy processes and reveal the more essential reason for the validity of (H) (see [21, Section 3] for examples).

Theorem 2.11

([21, Theorem 2.3]) Assume that all 1-excessive functions are lower semicontinuous.

(i) $X$ satisfies (H) if the following condition holds:

[TABLE]

(ii) Suppose $X$ satisfies (H). Then, for any finite measure $\nu$ on $\mathbf{R}^{n}$ of finite 1-energy and any $\varsigma>1$, there exists a sequence $\{y_{k}\uparrow\infty\}$ such that $y_{1}>1$ and (2.11) holds.

Let $\varsigma>1$ be a constant. We define $N^{\varsigma}_{x}:=\varsigma^{(\varsigma^{x})}\ \ {\rm for}\ x\geq 0.$

Theorem 2.12

([21, Theorem 2.4]) Assume that all 1-excessive functions are lower semicontinuous.

(i) $X$ satisfies (H) if the following condition holds:

[TABLE]

(ii) Suppose $X$ satisfies (H). Then, for any finite measure $\nu$ on $\mathbf{R}^{n}$ of finite 1-energy and any $\varsigma>1$, there exists a sequence of positive numbers $\{x_{k}\}$ such that $N^{\varsigma}_{x_{1}}>e$, $x_{k}+1<x_{k+1}$, $k\in\mathbf{N}$, $\sum_{k=1}^{\infty}\frac{1}{x_{k}}=\infty$, and (2.12) holds.

The proofs of the above two theorems rely on the following characterization for (H).

Proposition 2.13

([21, Proposition 2.2]) Assume that all 1-excessive functions are lower semicontinuous. Then (H) holds if and only if

[TABLE]

for any finite measure $\nu$ of finite 1-energy.

Motivated by exploration of Getoor’s conjecture for one-dimensional Lévy processes (see [22, Section 2.1]), we considered in [22] Hunt’s hypothesis (H) for the sum of two independent Lévy processes.

Theorem 2.14

([22, Theorem 3.1]) Let $X_{1}$ and $X_{2}$ be two independent Lévy processes on $\mathbf{R}^{n}$. If $X_{1}$ satisfies (H) and $X_{2}$ is a compound Poisson process, then $X_{1}+X_{2}$ satisfies (H).

Hereafter we say that a Lévy process with Lévy-Khintcine exponent $(a,Q,\mu)$ satisfies condition (S) if $\mu({\mathbf{R}^{n}\backslash\sqrt{Q}\mathbf{R}^{n}})<\infty$ and the equation $\sqrt{Q}y=-a-\int_{\{\mathbf{R}^{n}\backslash\sqrt{Q}\mathbf{R}^{n}\}}x1_{\{|x|<1\}}\mu(dx)$ has at least one solution $y\in\mathbf{R}^{n}$.

Theorem 2.15

([22, Theorem 3.2]) Let $X_{1}$ and $X_{2}$ be two independent Lévy processes on $\mathbf{R}^{n}$. If both $X_{1}$ and $X_{2}$ satisfy condition (S), then $X_{1}+X_{2}$ satisfies (H).

To show Theorem 2.14, we considered projections for Lévy processes (see [22, Lemma 3.4]) and used an idea in the proof of [20, Theorem 1.2] (see [22, Lemma 3.6]). To show Theorem 2.15, we proved a lemma for general symmetric nonnegative matrices (see [22, Lemma 3.7]).

If Lévy processes have resolvent densities, we have the following result on the validity of (H).

Theorem 2.16

([22, Theorem 4.1]) Assume that $X_{1}$ and $X_{2}$ are two independent Lévy processes on $\mathbf{R}^{n}$ such that $X_{1}+X_{2}$ has resolvent densities with respect to $m_{n}$. Denote by $\psi_{1}$ and $\psi_{2}$ the Lévy-Khintchine exponents of $X_{1}$ and $X_{2}$, respectively. Suppose that

(i) $X_{1}$ has resolvent densities with respect to $m_{n}$ and satisfies (H).

(ii) Any finite measure $\nu$ of finite 1-energy with respect to $X_{1}+X_{2}$ has finite 1-energy with respect to $X_{1}$.

*(iii) There exists a constant $c>0$ such that $|{\rm Im}(\psi_{2})|\leq c(1+{\rm Re}(\psi_{1})+{\rm Re}(\psi_{2})).$

Then $X_{1}+X_{2}$ satisfies (H).*

For condition (ii) of Theorem 2.16, we refer the reader to [22, Proposition 4.2] for some sufficient conditions.

Before ending this subsection, we present a result which implies that big jumps have no effect on the validity of (H) for any Lévy process.

Theorem 2.17

([23, Proposition 4.11]) Suppose that $\mu_{1}$ is a finite measure on $\mathbf{R}^{n}\backslash\{0\}$ such that $\mu_{1}\leq\mu$. Denote $\mu^{\prime}:=\mu-\mu_{1}$ and let $X^{\prime}$ be a Lévy process on $\mathbf{R}^{n}$ with Lévy-Khintchine exponent $(a^{\prime},Q,\mu^{\prime})$, where $a^{\prime}:=a+\int_{\{|x|<1\}}x\mu_{1}(dx).$ Then,

(i) $X$ and $X^{\prime}$ have same semipolar sets.

(ii) $X$ and $X^{\prime}$ have same $m_{n}$-essentially polar sets.

(iii) $X$ satisfies (H) if and only if $X^{\prime}$ satisfies (H).

(iv) $X$ satisfies $(H_{m_{n}})$ if and only if $X^{\prime}$ satisfies $(H_{m_{n}})$, where $(H_{m_{n}})$ means that every semipolar set is $m_{n}$-essentially polar.

We proved Theorem 2.17 (i) and (ii) in [24, Theorem 2.1]. (iv) is a direct consequence of (i) and (ii), and (iii) is based on [22, Theorem 3.1] and [23, Theorem 1.1, Corollary 4.10].

(2) (H) for one-dimensional Lévy processes

In this part, we assume that $X$ is a one-dimensional Lévy process with Lévy-Khintchine exponent $(a,Q,\mu)$. Denote by $\mu_{+}$ and $\mu_{-}$ the restriction of $\mu$ on $(0,\infty)$ and $(-\infty,0)$, respectively. Let $\bar{\mu}_{-}$ be the image measure of $\mu_{-}$ under the map $x\mapsto-x,\ \forall x\in(-\infty,0).$ The following result extends Kesten [29, Theorem 1(f)].

Theorem 2.18

([22, Theorem 2.2]) Suppose that $Q=0$ and $\int_{0}^{\infty}(1\wedge x)\mu_{+}(dx)=\infty$. If there exist $\delta\in(0,1),k\in[0,1)$, and a measure $\nu$ on $\mathbf{R}^{+}$ satisfying $\int_{0}^{\delta}x\nu(dx)<\infty$, such that $\bar{\mu}_{-}\leq k\mu_{+}+\nu.$ Then $X$ satisfies (H).

The basic idea of the proof for Theorem 2.18 is to use Kesten’s criterion (2.1) and Bretagnolle’s beautiful characterization of one-dimensional Lévy processes (see [5, Theorem 8]).

We now give a novel condition on the Lévy measure $\mu$ which implies (H) for a large class of one-dimensional Lévy processes.

Theorem 2.19

([22, Theorem 2.2]) If

[TABLE]

then $X$ satisfies (H).

The proof of Theorem 2.19 is based on the following characterization for (H).

Proposition 2.20

([22, Proposition 2.4]) Suppose that $X$ is a Lévy process on $\mathbf{R}^{n}$ which has resolvent densities with respect to $m_{n}$. Let $f$ be a positive increasing function on $[1,\infty)$ such that $\int_{N}^{\infty}(\lambda f(\lambda))^{-1}d\lambda=\infty$ for some $N\geq 1$. Then (H) holds for $X$ if and only if

[TABLE]

for any finite measure $\nu$ of finite 1-energy.

Remark 2.21

Note that, different from most existing sufficient conditions for (H), our condition (2.5) does not require any controllability of ${\rm Im}(\psi)$ by $1+{\rm Re}(\psi)$. Define the measure $\xi$ on $\mathbf{R}$ by

[TABLE]

Condition (2.5) is slightly stronger than $\xi$ is an infinite measure on $\mathbf{R}$. We refer the reader to [22, Remark 2.5] for more details.

From the proof of Theorem 2.19, we we can see that the following result extending [24, Theorem 4.7] holds.

Proposition 2.22

([22, Proposition 2.6]) If

[TABLE]

then $X$ satisfies (H).

Following the proof of Theorem 2.19, we can prove the following result.

Proposition 2.23

([22, Proposition 2.7]) If

[TABLE]

then $X$ satisfies (H).

(3) (H) for subordinators

$X$ is called a subordinator if it is a one-dimensional increasing Lévy process. Subordinators are a very important class of Lévy processes. Let $X$ be a subordinator. Then its Lévy-Khintchine exponent $\psi$ can be expressed by

[TABLE]

where $d\geq 0$ (called the drift coefficient) and $\mu$ satisfies $\int_{0}^{\infty}(1\wedge x)\mu(dx)<\infty$.

Proposition 2.24

([20, Proposition 1.6]) If $X$ is a subordinator and satisfies (H), then $d=0$.

Proposition 2.24 can be extended to the high-dimensional case, see Proposition 3.2 below.

A natural question is: if $X$ is a pure jump subordinator, i.e., $d=0$, must $X$ satisfy (H)? Up to now, it is still unknown if the answer is yes or no. In the following, we first show that some particular subordinators satisfy (H).

Recall that the potential measure $U$ of $X$ is defined by

[TABLE]

$X$ is called a special subordinator if $U|_{(0,\infty)}$ has a decreasing density with respect to the Lebesgue measure.

Theorem 2.25

([24, Theorem 3.3]) Let $X$ be a special subordinator. Then $X$ satisfies (H) if and only if $d=0$.

Definition 2.26

([24, Definition 3.4]) Let $X$ be a subordinator with drift [math] and Lévy measure $\mu$. We call $X$ a locally quasi-stable subordinator if there exist a stable subordinator $S$ with Lévy measure $\mu_{S}$, positive constants $c_{1},c_{2},\delta$, and finite measures $\mu_{1}$ and $\mu_{2}$ on $(0,\delta)$ such that

[TABLE]

Proposition 2.27

([24, Proposition 3.5]) Any locally quasi-stable subordinator satisfies (H).

We refer the reader to [24, Section 3.3] and [22, Example 4.10] for more examples on subordinators satisfying (H).

In [24, Section 5], we constructed a type of subordinators that does not satisfy Rao’s condition. So far we have not been able to prove or disprove that (H) holds for the subordinators. The example suggests that maybe completely new ideas and methods are needed for resolving Getoor’s conjecture.

Definition 2.28

([21, Definition 4.1]) Let $0<\alpha<\beta<1$. A pure jump subordinator $X$ is said to be of type-$(\alpha,\beta)$ if the Lévy measure of $X$ has density, which is denoted by $\rho$, and there exists a constant $c>1$ such that

[TABLE]

Up to now it is still unknown if any pure jump subordinator of type-$(\alpha,\beta)$ satisfies (H). But we have proved the following result based on Theorem 2.11.

Theorem 2.29

([21, Theorem 4.2]) Any pure jump subordinator of type-$(\alpha,\beta)$ can be decomposed into the summation of two independent pure jump subordinators of type-$(\alpha,\beta)$ such that both of them satisfy (H).

2.2 (H) for Markov processes

In this subsection, we assume that $E$ is a locally compact space with a countable base and $X=(X_{t},P^{x})$ is a standard Markov process on $E$.

Suppose that $X$ is associated with a (not necessarily symmetric) regular Dirichlet form on $L^{2}(E;m)$, where $m$ is a Radon measure on $E$. Silverstein [34] proved that any semipolar set for $X$ is $m$-essentially polar. This result plays a very important role in the theory of Dirichlet forms. For example, it is used to prove the relationship between orthogonal projections and hitting distributions (cf. [11, Theorem 4.3.1] and its proof). Fitzsimmons [7] extended the result to the semi-Dirichlet forms setting and Han et al. [16] extended it to the positivity-preserving forms setting.

In [15], Glover and Rao gave a sufficient condition for nonsymmetric Hunt processes to satisfy (H). In [8], Fitzsimmons showed that Gross’s Bwownian motion, which is an infinite-dimensional Lévy process, fails to satisfy (H). In [17], Hansen and Netuka showed that (H) holds if there exists a Green function $G>0$ which locally satisfies the triangle inequality $G(x,z)\wedge G(y,z)\leq CG(x,y)$, where $C$ is a positive constant.

In [23], we investigated the invariance of (H) for Markov processes under two classes of transformations, which are change of measure and subordination. Before stating our results, we give some notation. We fix an isolated point $\Delta$ which is not in $E$ and write $E_{\Delta}=E\cup\{\Delta\}$. Consider the following objects:

(i) $\Omega$ is a set and $\omega_{\Delta}$ is a distinguished point of $\Omega$.

(ii) For $0\leq t\leq\infty$, $Z_{t}:\Omega\rightarrow E_{\Delta}$ is a map such that if $Z_{t}(\omega)=\Delta$ then $Z_{s}(\omega)=\Delta$ for all $s\geq t$, $Z_{\infty}(\omega)=\Delta$ for all $\omega\in\Omega$, and $Z_{0}(\omega_{\Delta})=\Delta$.

(iii) For $0\leq t\leq\infty$, $\theta_{t}:\Omega\rightarrow\Omega$ is a map such that $Z_{s}\circ\theta_{t}=Z_{s+t}$ for all $s,t\in[0,\infty]$, and $\theta_{\infty}\omega=\omega_{\Delta}$ for all $\omega\in\Omega$.

We define in $\Omega$ the $\sigma$-algebras $\mathcal{F}^{0}=\sigma(Z_{t}:t\in[0,\infty])$ and $\mathcal{F}^{0}_{t}=\sigma(Z_{s}:s\leq t)$ for $0\leq t<\infty$. Denote

[TABLE]

Let $m$ be a measure on $(E,{\mathcal{B}})$. We define

[TABLE]

Note that if a standard process $X$ has resolvent densities with respect to $m$, then $X$ satisfies (H) if and only if $X$ satisfies $(H_{m})$ (cf. [2, Propositions II.2.8 and II.3.2]).

Theorem 2.30

([23, Theorem 1.1]) Let $X=(\Omega,\mathcal{M}^{X},\mathcal{M}^{X}_{t},Z_{t},\theta_{t},P^{x})$ and $Y=(\Omega,\mathcal{M}^{Y},\mathcal{M}^{Y}_{t},Z_{t},$ $\theta_{t},Q^{x})$ be two standard processes on $E$ such that $\mathcal{M}^{X}\cap\mathcal{M}^{Y}\supset\mathcal{F}^{0}$ and $\mathcal{M}^{X}_{t}\cap\mathcal{M}^{Y}_{t}\supset\mathcal{F}_{t}^{0}$ for $0\leq t<\infty$.

(i) Suppose that $X$ satisfies (H) and for any $x\in E$ and $t>0$, $Q^{x}|_{\mathcal{F}_{t}^{0}}$ is absolutely continuous with respect to $P^{x}|_{\mathcal{F}_{t}^{0}}$ on $\{t<\zeta\}$. Then $Y$ satisfies (H).

(ii) Suppose that $X$ satisfies ($H_{m}$) for some measure $m$ on $(E,{\mathcal{B}})$ and for any $x\in E$ and $t>0$, $Q^{x}|_{\mathcal{F}_{t}^{0}}$ is absolutely continuous with respect to $P^{x}|_{\mathcal{F}_{t}^{0}}$ on $\{t<\zeta\}$. Then $Y$ satisfies ($H_{m}$).

Let $X=(X_{t})_{t\geq 0}$ be a standard process on $E$ and $\tau$ be a subordinator which is independent of $X$. The standard process $(X_{\tau_{t}})_{t\geq 0}$ is called the subordinated process of $(X_{t})_{t\geq 0}$. The idea of subordination originated from Bochner (cf. [4]). Our next result is motivated by the following remarkable theorem of Glover and Rao.

Theorem 2.31

(Glover and Rao [14]) Let $(X_{t})_{t\geq 0}$ be a standard process on $E$ and $(\tau_{t})_{t\geq 0}$ be a subordinator which is independent of $X$ and satisfies (H). Then $(X_{\tau_{t}})_{t\geq 0}$ satisfies (H).

Now we present our result on the equivalence between (H) for $X$ and (H) for its time changed process.

Theorem 2.32

([23, Theorem 1.3]) Let $(X_{t})_{t\geq 0}$ be a standard process on $E$ and $m$ be a measure on $(E,{\mathcal{B}})$. Then,

(i) $(X_{t})_{t\geq 0}$ satisfies (H) if and only if $(X_{\tau_{t}})_{t\geq 0}$ satisfies (H) for some (and hence any) subordinator $(\tau_{t})_{t\geq 0}$ which is independent of $(X_{t})_{t\geq 0}$ and has a positive drift coefficient.

(ii) $(X_{t})_{t\geq 0}$ satisfies ($H_{m}$) if and only if $(X_{\tau_{t}})_{t\geq 0}$ satisfies ($H_{m}$) for some (and hence any) subordinator $(\tau_{t})_{t\geq 0}$ which is independent of $(X_{t})_{t\geq 0}$ and has a positive drift coefficient.

The proof of Theorem 2.30 is based on two lemmas ([23, Lemmas 2.1 and 2.2]) and Blumenthal’s 0-1 law. The proof of Theorem 2.32 is based on [23, Lemma 2.1] and Bertoin [1, Theorem III.5]. We refer the reader to [23, Theorems 4.3, 4.7, 4.13 and Proposition 5.1] for applications of Theorems 2.30 and 2.32.

3 (H) for multidimensional Lévy processes: projections

In this section, we investigate (H) for multidimensional Lévy processes from the viewpoint of projections. Throughout this section, we assume that $n>1$ except in Proposition 3.2 below and $X=(X_{t})_{t\geq 0}$ is a Lévy process on $\mathbf{R}^{n}$ with Lévy-Khintchine exponent $(a,Q,\mu)$. For a subspace $A$ of $\mathbf{R}^{n}$, we use $A^{\perp}$ to denote its orthogonal complement space.

3.1 A lemma on projections and applications

In [22], we proved the following result.

Lemma 3.1

([22, Lemma 3.4]) Suppose that $X$ satisfies (H). Then for any nonempty proper subspace $A$ of $\mathbf{R}^{n}$, the projection process $Y=(Y_{t})_{t\geq 0}$ of $X$ on $A$ satisfies (H).

As an application of Lemma 3.1, we proved in [22] Theorem 2.14 of Section 2. As another application of Lemma 3.1, we proved in [23] the following result, which extends Proposition 2.24.

Proposition 3.2

([23, Proposition 5.3]) Let $X$ be a Lévy process on $\mathbf{R}^{n}\,(n\geq 1)$ with Lévy-Khintchine exponent $(a,0,\mu)$ satisfying $\int_{\mathbf{R}^{n}}(|x|\wedge 1)\mu(dx)<\infty$. If $X$ satisfies (H), then its drift coefficient equals zero.

Proposition 3.2 can be further extended as follows.

Proposition 3.3

Suppose that $Q$ is degenerate. Let $Y=(Y_{t})_{t\geq 0}$ be the projection process of $X$ on $(\sqrt{Q}\mathbf{R}^{n})^{\perp}$. Denote by $P_{2}$ the projection operator from $\mathbf{R}^{n}$ to $(\sqrt{Q}\mathbf{R}^{n})^{\perp}$ and denote by $\mu_{P_{2}}$ the image measure of $\mu$ under $P_{2}$. Assume that

[TABLE]

If $X$ satisfies (H), then the drift coefficient of $Y$ equals zero.

Proposition 3.3 is a direct consequence of Lemma 3.1, Proposition 3.2 and the following lemma.

Lemma 3.4

Suppose that $Q$ is degenerate. Let $Y=(Y_{t})_{t\geq 0}$ be the projection process of $X$ on $(\sqrt{Q}\mathbf{R}^{n})^{\perp}$. Denote by $P_{2}$ the projection operator from $\mathbf{R}^{n}$ to $(\sqrt{Q}\mathbf{R}^{n})^{\perp}$ and denote by $\mu_{P_{2}}$ the image measure of $\mu$ under $P_{2}$. Define

[TABLE]

Then the Lévy-Khintchine exponent of $Y$ is $(a^{\prime},0,\mu_{P_{2}})$.

Proof. We use $k$ to denote the rank of $Q$ and assume without loss of generality that $1\leq k<n$. Then there exists an orthogonal matrix $O$ such that

[TABLE]

where $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{k}>0$, $\lambda_{i}=0$ for $i=k+1,\dots,n$, and $O^{T}$ denotes the transpose of $O$. Define $U=O^{T}diag(\sqrt{\lambda_{1}},\dots,\sqrt{\lambda_{n}})O$. Then $Q=U^{2}$ and hence $\sqrt{Q}=U$.

Let $E_{k}=diag(1,\dots,1,0,\dots,0)$ be the diagonal matrix with the first $k$ elements being 1. Then, we have that

[TABLE]

Define

[TABLE]

Then (3.2) implies that $P_{1}$ is the projection operator from $\mathbf{R}^{n}$ to $\sqrt{Q}\mathbf{R}^{n}$. Define

[TABLE]

where $I_{n}$ is the identity operator on $\mathbf{R}^{n}$. Then, $P_{2}$ is the projection operator from $\mathbf{R}^{n}$ to $(\sqrt{Q}\mathbf{R}^{n})^{\perp}$.

Now we compute the Lévy-Khintchine exponent of $Y$. Note that $Y_{t}=P_{2}X_{t}$ for $t\geq 0$. For $z\in\mathbf{R}^{n}$, we have

[TABLE]

By (3.1) and (3.3), we get

[TABLE]

Therefore, the Lévy-Khintchine exponent of $Y$ is $(a^{\prime},0,\mu_{P_{2}})$.

3.2 Converse of Lemma 3.1

In this subsection, we consider the converse of Lemma 3.1. We are particularly interested in the following questions:

Question 1. If for any nonempty proper subspace $A$ of $\mathbf{R}^{n}$, the projection process $X_{A}$ of $X$ on $A$ satisfies (H), does $X$ satisfy (H)?

Question 2. If for any one-dimensional subspace $A$ of $\mathbf{R}^{n}$, the projection process $X_{A}$ of $X$ on $A$ satisfies (H), does $X$ satisfy (H)?

Question 3. If the one-dimensional projection process of $X$ on each coordinate-axis satisfies (H), does $X$ satisfy (H)?

Question 4. Let $S$ be a nonempty proper subspace of $\mathbf{R}^{n}$. Assume that the two projection processes $X_{S}$ and $X_{S^{\perp}}$ of $X$ on $S$ and $S^{\perp}$, respectively, are independent and satisfy (H). Does $X$ satisfy (H)?

3.2.1 Counterexample for Question 3

We use a counterexample to show that the answer to Question 3 is negative.

Example 3.5

Let $B=(B_{t})_{t\geq 0}$ be a standard two-dimensional Brownian motion. Define

[TABLE]

Then,

[TABLE]

and $\sqrt{Q}\mathbf{R}^{2}=\{(x,y)\in\mathbf{R}^{2}:x=y\}$.

Let $a=(1,-1)^{T}$ and define $X=(X_{t})_{t\geq 0}$ by

[TABLE]

Then, the projection process of $X$ on each coordinate-axis has non-degenerate Gaussian part and thus satisfies (H) by Theorem 2.5. On the other hand, the projection process of $X$ on the subspace $\{(x,y)\in\mathbf{R}^{2}:y=-x\}$ is the uniform motion, which does not satisfy (H). Therefore, $X$ does not satisfy (H) by Lemma 3.1. Note that in this example the projection process of $X$ on any one-dimensional subspace of $\mathbf{R}^{2}$, except for $\{(x,y)\in\mathbf{R}^{2}:y=-x\}$, satisfies (H).

3.2.2 Partial answer to Question 2

In this part, we give an affirmative answer to Question 2 under the assumption that $\mu(\mathbf{R}^{n}$ $\backslash\sqrt{Q}\mathbf{R}^{n})<\infty$.

Theorem 3.6

Suppose that $Q$ is degenerate and $\mu(\mathbf{R}^{n}\backslash\sqrt{Q}\mathbf{R}^{n})<\infty$. Then the following three claims are equivalent:

(i) $X$ satisfies (H);

(ii) for any one-dimensional subspace $A$ of $\mathbf{R}^{n}$, the projection process of $X$ on $A$ satisfies (H);

(iii) the projection process of $X$ on $(\sqrt{Q}\mathbf{R}^{n})^{\bot}$ satisfies (H).

Proof. $(i)\Rightarrow(ii)$ and $(iii)$: This is a direct consequence of Lemma 3.1.

$(ii)\Rightarrow(i)$: Let $k$ be the rank of $Q$. Then $0\leq k<n$. By the orthogonal transformation of Lévy processes (cf. [20, Section 2.2]), we can assume without loss of generality that $Q={\rm diag}(\lambda_{1},\ldots,\lambda_{n})$, where $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{k}>0,\lambda_{k+1}=\cdots=\lambda_{n}=0$, and $X$ has the expression

[TABLE]

where

[TABLE]

where $b^{\prime}=-a-\int_{\{|x|<1\}}\mu_{1}(dx)$, $\mu_{1}$ is the restriction of $\mu$ on $\mathbf{R}^{n}\backslash\sqrt{Q}\mathbf{R}^{n}=\mathbf{R}^{k}\times(\mathbf{R}^{n-k}\backslash\{0\})$, $N$ is a Poisson random measure on $\mathbf{R}^{+}\times(\mathbf{R}^{n}\backslash\{0\})$ which is independent of the standard Brownian motion $B=(B_{t})_{t\geq 0}$, and $\tilde{N}(t,F)=N(t,F)-t\mu(F)$.

By the assumption $\mu(\mathbf{R}^{n}\backslash\sqrt{Q}\mathbf{R}^{n})<\infty$, we find that $(X^{(2)}_{t})_{t\geq 0}$ is a compound Poisson process. Denote $b^{\prime}=(b^{\prime}_{1},\ldots,b^{\prime}_{k},b^{\prime}_{k+1},\ldots,b^{\prime}_{n})$ and let ${\mathcal{P}}_{i}$ be the projection of $\mathbf{R}^{n}$ on the one-dimensional subspace $A_{i}:=\{(x_{1},\ldots,x_{n})\in\mathbf{R}^{n}:x_{j}=0,\ \forall j\neq i\}$ for $i=k+1,\ldots,n$. Then we have that

[TABLE]

where $({\mathcal{P}}_{i}X_{t}^{(2)})_{t\geq 0}$ is a compound Poisson process and ${\mathcal{P}}_{i}b^{\prime}\in A_{i}$.

By (ii), we find that the projection process $({\mathcal{P}}_{i}X_{t})_{t\geq 0}$ satisfies (H) and hence $b^{\prime}_{i}=0$ for any $i=k+1,\dots,n$. Then $b^{\prime}\in\mathbf{R}^{k}\times\{0\}=\sqrt{Q}\mathbf{R}^{n}$. Therefore, $X$ satisfies (H) by Theorem 2.6.

$(iii)\Rightarrow(i)$: Denote by $P_{2}$ the projection operator from $\mathbf{R}^{n}$ to $(\sqrt{Q}\mathbf{R}^{n})^{\perp}$. By (3.6), we get

[TABLE]

where $(P_{2}X_{t}^{(2)})_{t\geq 0}$ is a compound Poisson process and $P_{2}b^{\prime}\in(\sqrt{Q}\mathbf{R}^{n})^{\bot}$. By (iii), we find that $(P_{2}X_{t})_{t\geq 0}$ satisfies (H) and hence $P_{2}b^{\prime}=0$, which implies that $b^{\prime}\in\sqrt{Q}\mathbf{R}^{n}$. Therefore, $X$ satisfies (H) by Theorem 2.6.

3.2.3 A result on Question 1

In this part, we give a partial result on Question 1.

Theorem 3.7

Suppose there exists a subspace $S$ of $\mathbf{R}^{n}$ such that $\sqrt{Q}\mathbf{R}^{n}\subsetneq S\subsetneq\mathbf{R}^{n}$ and $\mu(\mathbf{R}^{n}\backslash S)<\infty$. Then the following three claims are equivalent:

(i) $X$ satisfies (H);

(ii) for any nonempty proper subspace $A$ of $\mathbf{R}^{n}$, the projection process of $X$ on $A$ satisfies (H);

(iii) for $A\in\{S,S^{\bot}\}$, the projection process of $X$ on $A$ satisfies (H).

Proof. By Lemma 3.1, we get $(i)\Rightarrow(ii)\Rightarrow(iii)$.

$(iii)\Rightarrow(i)$: Let $k$ be the dimension of $S$. Then $0<k<n$. By the orthogonal transformation of Lévy processes, we can assume without loss of generality that $S=\{(x_{1},\dots,x_{n})\in\mathbf{R}^{n}:x_{k+1}=\cdots=x_{n}=0\}$. By the assumption that $\sqrt{Q}\mathbf{R}^{n}\subsetneq S$, we can express $X$ as

[TABLE]

where

[TABLE]

where $b^{\prime}=-a-\int_{\{|x|<1\}}\mu_{S}(dx)$, $\mu_{S}$ is the restriction of $\mu$ on $\mathbf{R}^{n}\backslash S=\mathbf{R}^{k}\times(\mathbf{R}^{n-k}\backslash\{0\})$, $N$ is a Poisson random measure on $\mathbf{R}^{+}\times(\mathbf{R}^{n}\backslash\{0\})$ which is independent of the standard Brownian motion $B=(B_{t})_{t\geq 0}$, and $\tilde{N}(t,F)=N(t,F)-t\mu(F)$.

By the assumption $\mu(\mathbf{R}^{n}\backslash S)<\infty$, we find that $(X^{(2)}_{t})_{t\geq 0}$ is a compound Poisson process. Denote $b^{\prime}=(b^{\prime}_{1},\ldots,b^{\prime}_{k},b^{\prime}_{k+1},\ldots,b^{\prime}_{n})$. Let ${\mathcal{P}}_{S}$ and ${\mathcal{P}}_{S^{\bot}}$ be the projections of $\mathbf{R}^{n}$ on the subspaces $S$ and $S^{\bot}$, respectively. Then we have that

[TABLE]

where $({\mathcal{P}}_{S^{\bot}}X_{t}^{(2)})_{t\geq 0}$ is a compound Poisson process and ${\mathcal{P}}_{S^{\bot}}b^{\prime}\in{S^{\bot}}$.

By (iii), we find that the projection process $({\mathcal{P}}_{S^{\bot}}X_{t})_{t\geq 0}$ satisfies (H) and hence ${\mathcal{P}}_{S^{\bot}}b^{\prime}=0$. Then $b^{\prime}\in\mathbf{R}^{k}\times\{0\}=S$. Thus

[TABLE]

which implies that

[TABLE]

where $(P_{S}X^{(2)}_{t})_{t\geq 0}$ is a compound Poisson process. By (iii) and Theorem 2.17 (iii), we conclude that $(X^{(1)}_{t})_{t\geq 0}$ satisfies (H).

Suppose that $F$ is a semipolar set of $X$. Note that $(X^{(1)}_{t})_{t\geq 0}$ satisfies (H), $(X^{(2)}_{t})_{t\geq 0}$ is a compound Poisson process, $(X^{(1)}_{t})_{t\geq 0}$ and $(X^{(2)}_{t})_{t\geq 0}$ are independent. Following the proof of [20, Theorem 1.2, $(ii)\Rightarrow(i)$], we can show that $F$ is a polar set of $X$ by (3.7). Therefore, $X$ satisfies (H).

Remark 3.8

In view of Theorem 3.7, we point out that the additional assumption that the projection process of $X$ on $S^{\bot}$ satisfies (H) should be added to [22, Lemma 3.5] in order that its conclusion holds. But [22, Lemma 3.6] is still true and thus [22, Theorem 3.1] (i.e., Theorem 2.14 of Section 2) holds.

3.2.4 Two propositions on Question 4

By the orthogonal transformation of Lévy processes, we find that Question 4 is equivalent to the following question:

Question 4’. Let $X_{1}$ and $X_{2}$ be Lévy processes on $\mathbf{R}^{n}$ and $\mathbf{R}^{m}$, respectively. Suppose that $X_{1}$ and $X_{2}$ are independent and both of them satisfy (H). Does the $\mathbf{R}^{n+m}$-valued Lévy process $X=(X_{1},X_{2})$ satisfy (H)?

Denote by $\psi_{1}$ and $(a_{1},Q_{1},\mu_{1})$ the Lévy-Khintchine exponent of $X_{1}$, by $\psi_{2}$ and $(a_{2},Q_{2},\mu_{2})$ the Lévy-Khintchine exponent of $X_{2}$, and by $\psi$ and $(a,Q,\mu)$ the Lévy-Khintchine exponent of $X$. Define the $(n+m)\times(n+m)$ matrix

[TABLE]

and two measures $\bar{\mu}_{1}$ and $\bar{\mu}_{2}$ on $\mathbf{R}^{n+m}$ by

[TABLE]

where $0_{n}$ and $0_{m}$ are zero elements of $\mathbf{R}^{n}$ and $\mathbf{R}^{m}$, respectively, and $A$ is an arbitrary Borel subset of $\mathbf{R}^{n+m}$. By direct calculation, we get

[TABLE]

and

[TABLE]

Proposition 3.9

([22, Lemma 3.6]) If $X_{1}$ satisfies (H) and $X_{2}$ is a compound Poisson process, then $X=(X_{1},X_{2})$ satisfies (H).

Proposition 3.10

If both $X_{1}$ and $X_{2}$ satisfy (S), then $X=(X_{1},X_{2})$ satisfies (H).

Proof. By the assumption we have that

[TABLE]

which together with (3.10), (3.11) and (3.13) implies that

[TABLE]

By Theorem 2.6, we find that both $X_{1}$ and $X_{2}$ satisfy the Kanda-Forst condition. Then $X$ satisfies the Kanda-Forst condition by (3.12). Therefore, $X=(X_{1},X_{2})$ satisfies (H) by Theorem 2.6 and (3.14).

4 Energy for multidimensional Lévy processes

Let $X=(X_{t})_{t\geq 0}$ be an $\mathbf{R}^{n}$-valued Lévy process on $(\Omega,{\mathcal{F}},P)$ with Lévy-Khintchine exponent $\psi$. For a finite measure $\nu$ on $\mathbf{R}^{n}$ and $\lambda>0$, we define its $\lambda$-energy $E_{X}^{\lambda}(\nu)$ by

[TABLE]

Energy plays a fundamental role in the study of Hunt’s hypothesis (H). Kanda and Rao gave the following remarkable result on the relation between a measure which does not charge semipolar sets and its energy.

Theorem 4.1

(Kanda [27] and Rao [33]) Assume that $X$ has resolvent densities with respect to $m_{n}$. Let $\nu$ be a finite measure which charges no semipolar sets and $E_{X}^{\lambda}(\nu)<\infty$ for $\lambda>0$. Then

[TABLE]

Based on Rao [33], we proved the following result.

Theorem 4.2

([24, Theorem 5.1]) Assume that $X$ has resolvent densities with respect to $m_{n}$. Then (H) holds if and only if

[TABLE]

for any finite measure $\nu$ of finite 1-energy.

Kanda considered in [28] the space-time process $Y=(Y_{t})_{t\geq 0}$ over $X$, which means that $Y$ is a Lévy process on $\mathbf{R}^{1}\times\mathbf{R}^{n}$ defined on the probability space $(\mathbf{R}^{1}\times\Omega,P^{r,x})$, where $P^{r,x}=\delta_{r}\otimes P^{x}$ with $\delta_{r}$ being the Dirac measure at $r\in\mathbf{R}^{1}$. The trajectory $Y_{t}(r,\omega)$ is $(r+t,X_{t}(\omega))$ and the Lévy-Khintchine exponent of $Y$ is $\psi(z)-it$.

Theorem 4.3

([28, Theorem]) Let $X$ be a Lévy process on $\mathbf{R}^{n}$ with transition probability densities and $Y$ be the space-time process over $X$. Let $\nu$ be a finite measure on $\mathbf{R}^{1}\times\mathbf{R}^{n}$ of compact support.

(I) Assume that the $\lambda$-energy of $\nu$ for $Y$ is finite. Then,

(i) The $\mathbf{R}^{n}$-marginal $\nu_{2}$ of $\nu$ (i.e., $\nu_{2}(B)=\nu(\mathbf{R}^{1}\times B)$) has finite $\lambda$-energy for $X$.

(ii) If the $\mathbf{R}^{1}$-marginal $\nu_{1}$ of $\nu$ (i.e., $\nu_{1}(B)=\nu(B\times\mathbf{R}^{n})$) is singular to the Lebesgue measure on $\mathbf{R}^{1}$, then the $\mathbf{R}^{n}$-marginal $\nu_{2}$ does not charge any semipolar set.

(II) Consider the case that $\nu$ is of the direct product form $\eta\otimes\xi$.

(i) If $\nu$ has finite $\lambda$-energy for $Y$ and $\xi$ is carried by a semipolar set for $X$, then $\eta$ has a $L^{2}$-density relative to the Lebesgue measure on $\mathbf{R}^{1}$.

(ii) If $\xi$ is a finite measure of compact support on $\mathbf{R}^{n}$ with finite $\lambda$-energy for $X$ and it does not charge any semipolar set for $X$, then we can find a singular measure $\eta$ of compact support so that $\nu=\eta\otimes\xi$ has finite $\lambda$-energy for $Y$.

Using Theorem 4.3, Kanda obtained the following characterization of semipolar sets.

Corollary 4.4

([28, Corollary]) Let $X$ be a Lévy process on $\mathbf{R}^{n}$ which has transition probability densities. Then a closed set $B$ in $\mathbf{R}^{n}$ is semipolar if and only if

[TABLE]

for every $x\in\mathbf{R}^{n}$ and every set $A\subset(0,\infty)$ of Lebesgue measure 0.

In this section, we will follow the idea of Kanda [28] to consider energy for multidimensional Lévy processes. From now on till the end of this section, we let $X=(X_{t})_{t\geq 0}$ be an $\mathbf{R}^{n}$-valued Lévy process and $Y=(Y_{t})_{t\geq 0}$ be an $\mathbf{R}^{m}$-valued Lévy process. Define $Z_{t}=(X_{t},Y_{t})$. Assume that $X$ and $Y$ are independent, $X$, $Y$ and $Z$ have resolvent densities relative to Lebesgue measures on $\mathbf{R}^{n},\mathbf{R}^{m}$ and $\mathbf{R}^{n+m}$, respectively. Denote the Lévy-Khintchine exponents of $X$ and $Y$ by $\Phi$ and $\Psi$, respectively. Then the exponent of $Z$ is $\Phi(x)+\Psi(y)$ for $x\in\mathbf{R}^{n}$ and $y\in\mathbf{R}^{m}$.

4.1 Results

Proposition 4.5

Suppose that $\nu$ is a finite measure on $\mathbf{R}^{n}\times\mathbf{R}^{m}$ of compact support and has finite $\lambda$-energy for $Z$. Then

(i) the $\mathbf{R}^{n}$-marginal $\nu_{1}$ of $\nu$ has finite $\lambda$-energy for $X$;

(ii) the $\mathbf{R}^{m}$-marginal $\nu_{2}$ of $\nu$ has finite $\lambda$-energy for $Y$.

Proposition 4.6

Suppose that $\nu$ is a finite measure on $\mathbf{R}^{n}\times\mathbf{R}^{m}$ of compact support with the direct product form $\eta\otimes\xi$ and has finite $\lambda$-energy for $Z$.

(i) If $\xi$ is carried by a semipolar set for $Y$, then $\eta$ has a $L^{2}$-density relative to the Lebesgue measure on $\mathbf{R}^{n}$;

(ii) If $\eta$ is carried by a semipolar set for $X$, then $\xi$ has a $L^{2}$-density relative to the Lebesgue measure on $\mathbf{R}^{m}$.

As a direct consequence of Proposition 4.6 and [28, Corollary of Lemma 2.1], we obtain the following result.

Corollary 4.7

Suppose that $\nu$ is a finite measure on $\mathbf{R}^{n}\times\mathbf{R}^{m}$ of compact support with the direct product form $\eta\otimes\xi$ and has finite $\lambda$-energy for $Z$.

(i) If $\eta$ is singular to the Lebesgue measure on $\mathbf{R}^{n}$, then $\xi$ does not charge any semipolar set;

(ii) If $\xi$ is singular to the Lebesgue measure on $\mathbf{R}^{m}$, then $\eta$ does not charge any semipolar set.

Proposition 4.8

Assume that $X$ satisfies (H) and $Y$ satisfies the Kanda-Forst condition. Suppose that $\nu$ is a finite measure on $\mathbf{R}^{n}\times\mathbf{R}^{m}$ of compact support with the direct product form $\eta\otimes\xi$ and has finite $\lambda$-energy for $Z$. Then,

[TABLE]

4.2 Proofs

Proof of Proposition 4.5. We only prove (i). The proof of (ii) is similar and we omit it here. We assume without loss of generality that $\nu$ is a probability measure on $\mathbf{R}^{n}\times\mathbf{R}^{m}$. Then $\nu$ can be disintegrated as

[TABLE]

where $\nu_{1}(dx)=\nu(dx\times\mathbf{R}^{m})$ and $\nu_{2}(x,dy)$ are probability measures on $\mathbf{R}^{n}$ and $\mathbf{R}^{m}$, respectively. Set

[TABLE]

Then

[TABLE]

By the assumption that the $\lambda$-energy of $\nu$ for $Z$ is finite, we get

[TABLE]

It follows that

[TABLE]

By (4.1), we get

[TABLE]

For $x\in\mathbf{R}^{n}$ and $y\in\mathbf{R}^{m}$, we have

[TABLE]

Then, we obtain by (4.2)–(4.2) that

[TABLE]

We have

[TABLE]

where

[TABLE]

It follows that $G_{1}(x,y)=G_{1}(-x,y)$, $G_{2}(x,y)=-G_{2}(-x,y)$, which together with ${\rm Re}([\lambda+\Phi(x)]^{-1})={\rm Re}([\lambda+\Phi(-x)]^{-1})$, (4.6) and (4.5) implies that for any $\varsigma>0$,

[TABLE]

Thus $E_{X}^{\lambda}({\rm Re}f(\cdot,y)\nu_{1})<\infty$ for almost all $y$, where

[TABLE]

By (4.7) and the assumption that $\nu$ has compact support, we find that there exist constants $c>0$ and $\varepsilon>0$ such that ${\rm Re}f(x,y)>c$ for every $|y|<\varepsilon$ and all $x\in{\mathbf{R}^{n}}$. Therefore, $E^{\lambda}_{X}(\nu_{1})<\infty$ by [28, Corollary of Lemma 2.1].

Proof of Proposition 4.6. We only prove (i). By Proposition 4.5, we get $E_{Y}^{\lambda}(\xi)<\infty$. If $\xi$ charges a semipolar set, then it charges a compact set $K\subset\mathbf{R}^{m}$ such that $K\subset\{y\in\mathbf{R}^{m}:E^{y}[\exp(-\lambda T_{K})]<\delta\}$ for some $\delta\in(0,1)$. Let $\xi_{K}$ be the restriction of $\xi$ on $K$. By [28, Corollary of Lemma 2.1], we have that $E^{\lambda}_{Y}(\xi_{K})\leq E^{\lambda}_{Y}(\xi)<\infty$. Then $K$ must be non-polar for $Y$ by [28, Lemma 2.3]. Hence $C^{\lambda}_{Y}(K)\uparrow C$ as $\lambda\uparrow\infty$ for some positive finite constant $C$ by [28, Lemma 2.4], where $C^{\lambda}_{Y}(K)$ is the $\lambda$-capacity of $K$ relative to the process $Y$. Thus, we obtain by [28, Lemma 2.2] that

[TABLE]

i.e.,

[TABLE]

Similar to (4.2), we can show that for $x\in\mathbf{R}^{n}$ and $y\in\mathbf{R}^{m}$,

[TABLE]

It follows from (4.8) and (4.9) that

[TABLE]

By Fatou’s lemma, we get

[TABLE]

Therefore, $\hat{\eta}\in L^{2}(\mathbf{R}^{n})$, which implies that $\eta$ is absolutely continuous relative to the Lebesgue measure on $\mathbf{R}^{n}$ and its density belongs to $L^{2}(\mathbf{R}^{n})$.

Proof of Proposition 4.8. By the assumption that $Y$ satisfies the Kanda-Forst condition and the proof of [22, Lemma 4.6], we find that there exist two positive constant $c_{1}$ and $c_{2}$ such that for any $y\in\mathbf{R}^{m}$ and any $\lambda>1$,

[TABLE]

and

[TABLE]

By Proposition 4.5, we know that $\eta$ has finite energy for $X$. Then, by the assumption that $X$ satisfies (H) and Theorem 4.2, we get

[TABLE]

By (4.10), Fatou’s lemma (based on (4.2), the monotonicity of $E^{\lambda}_{X}(\eta)$ and the assumption that $\nu$ has finite $\lambda$-energy for $Z$), and (4.12), we get

[TABLE]

The proof is complete.

5 Open questions

Besides Questions 1, 2, and 4 (equivalently, 4’) given in §3.2, we would like to present a few more questions on Hunt’s hypothesis (H) for further study.

Question 5. Does any pure jump subordinator with infinite Lévy measure satisfy (H)? More generally, does any pure jump $n$-dimensional Lévy process satisfy (H)? The latter question has been presented in [23].

Question 6. Dose any one-dimensional Lévy process with $\int_{\mathbf{R}}(1\wedge|x|)\mu(dx)=\infty$ satisfy (H)?

Question 7. Suppose that $X$ and $Y$ are two independent $n$-dimensional Lévy processes satisfying (H). Does $X+Y$ satisfy (H)? We have mentioned this question in [21].

Remark 5.1

(i) By Kesten [29, Theorem 1(f)], we know that if the answer to Question 7 is yes, then the answer to Question 6 is also yes (see [22, Section 2.1]).

(ii) By Lemma 3.1 and the orthogonal transformation of Lévy processes, we know that if the answer to Question 4 is yes, then the answer to Question 7 is also yes.

Acknowledgments This work was supported by National Natural Science Foundation of China (Grant No. 11771309 and No. 11871184), Natural Science and Engineering Research Council of Canada.