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Further study on Hunt's hypothesis (H) for Levy processes

Getoor's conjecture that essentially all Levy processes satisfy (H) is a long-standing open problem in potential theory. In the beginning of the paper, we summarize the main results obtained so far for the problem. Then, we present two new necessary and sufficient conditions for the validity of (H). Furthermore, we give applications of these new criteria. First, we give explicit constructions of Levy processes satisfying (H) in a context where previously known results could not be applied. Second, we show that a large class of pure jump subordinators can be decomposed into the summation of two independent subordinators such that both of them satisfy (H).


Introduction
A Lévy process X on R n is said to satisfy Hunt's hypothesis (H) if every semipolar set of X is polar. More than forty years ago, Professor R.K. Getoor raised the problem that for which Lévy processes semipolar sets are always polar. He conjectured that essentially all Lévy processes satisfy (H). Getoor's conjecture is the major open problem in the field of potential theory for Lévy processes (cf. e.g. [1, page 70]). In this paper, we will present new criteria for the validity of (H).
Let us start with a brief introduction to Hunt's hypothesis (H). For simplicity, we consider here only (H) for Lévy processes; however, we should point out that (H) plays a crucial role in the potential theory of (dual) Markov processes. We refer the readers to [3, Chapter VI] for a systematic introduction to (H) for Markov processes.
Hereafter E denotes the expectation with respect to P and ·, · denotes the Euclidean inner product of R n . The classical Lévy-Khintchine formula tells us that where a ∈ R n , Q is a symmetric nonnegative definite n × n matrix, and µ is a measure (called the Lévy measure) on R n \{0} satisfying R n \{0} (1 ∧ |x| 2 )µ(dx) < ∞. We use Re(ψ) and Im(ψ) to denote respectively the real and imaginary parts of ψ, and use (a, Q, µ) to denote ψ. For x ∈ R n , we denote by P x the law of x + X under P . In particular, P 0 = P .
Denote by B n the family of all nearly Borel measurable subsets of R n . For D ⊂ E, we denote the first hitting time of D by σ D := inf{t > 0 : X t ∈ D}.
A set D ⊂ E is called polar if there exists a set C ∈ B n such that D ⊂ C and P x (σ C < ∞) = 0 for every x ∈ R n . D is called a thin set if there exists a set C ∈ B n such that D ⊂ C and P x (σ C = 0) = 0 for every x ∈ R n . D is called semipolar if D ⊂ ∞ n=1 D n for some thin sets {D n } ∞ n=1 . X is said to satisfy Hunt's hypothesis (H) if every semipolar set of X is polar.
To appreciate the importance of (H), we recall below some important principles of potential theory that are equivalent to (H). We refer the readers to [11,Proposition 1.1] for a summary of the proofs. For α > 0, a finite α-excessive function f on R n is called a regular potential provided that E x {e −αTn f (X Tn )} → E x {e −αT f (X T )} for x ∈ E whenever {T n } is an increasing sequence of stopping times with limit T . Denote by (U α ) α>0 the resolvent operators for X.
• Bounded energy principle: If ν is a finite measure with compact support such that U α ν is bounded, then ν does not charge semipolar sets.
• Bounded regularity principle: If ν is a finite measure with compact support such that U α ν is bounded, then U α ν is regular.
Hunt's hypothesis (H) is also equivalent to some other important properties of Markov processes. For example, Blumenthal and Getoor [4, Proposition (4.1)] and Glover [7,Theorem (2.2)] showed that (H) holds if and only if the fine and cofine topologies differ by polar sets; Fitzsimmons and Kanda [5] showed that (H) is equivalent to the dichotomy of capacity.
In spite of its importance, Hunt's hypothesis (H) has been verified only in special situations. Some forty years ago, Getoor conjectured that essentially all Lévy processes satisfy (H). To motivate this paper, we recall below the results obtained so far for Getoor's conjecture that are closely related to our results.
Blumenthal and Getoor [4] showed that all stable processes with index α ∈ (0, 2) on the line satisfy (H). Kanda [12] and Forst [6] proved independently that (H) holds if X has bounded continuous transition densities (with respect to the Lebesgue measure dx) and the Lévy-Khintchine exponent ψ satisfies for some constant M > 0. Rao [14] gave a short proof of the Kanda-Forst theorem under the weaker condition that X has resolvent densities. In particular, for n ≥ 1, all stable processes with index α = 1 satisfy (H). Kanda [13] proved that (H) holds for stable processes on R n with index α = 1 if we assume that the linear term vanishes. Glover and Rao [8] proved that α-subordinates of general Hunt processes satisfy (H) (cf. Proposition 3.3 below). Rao [15] proved that if all 1excessive functions of X are lower semicontinuous and Hu and Sun [10] showed that if Q is non-degenerate then X satisfies (H); if Q is degenerate then, under the assumption that µ(R n \ √ QR n ) < ∞, X satisfies (H) if and only if the equation has at least one solution y ∈ R n . They also showed that if X is a subordinator satisfying (H) then its drift coefficient must be 0. Recently Hu, Sun and Zhang [11] gave a comparison result on Lévy processes which implies that big jumps have no effect on the validity of (H). Also, they gave a new necessary and sufficient condition for (H) (cf. Proposition 2.1 below) and obtained an extended Kanda-Forst-Rao theorem.
In the next section of this paper, we will present novel necessary and sufficient conditions for the validity of (H) for Lévy processes. It is a bit surprising to us that these criteria were not known until now. Based on these criteria, we will give in Section 3 new examples of Lévy processes satisfying (H).

New criteria for Hunt's hypothesis (H)
Let X be a Lévy process on R n with Lévy-Khintchine exponent ψ. From now on we assume that all 1-excessive functions are lower semicontinuous, equivalently, X has resolvent densities. We refer the readers to [9, Theorem 2.1] for more characterizations of this weak assumption.
For a finite (positive) measure ν on R n , we denotê ν is said to have finite 1-energy if Let ν be a finite measure on R n of finite 1-energy. For λ > 0, define Then, we have (see [15, Theorem 1 and the proof of The for any finite measure ν of finite 1-energy. For ς > 1, we define N ς x := ς (ς x ) for x ≥ 0. To simplify notations, we use N to denote N ς whenever there is no confusion caused. Throughout this paper, we use log to denote log e . Theorem 2.2. Let ν be a finite measure on R n of finite 1-energy. Then, the following condition is fulfilled for any δ > 0.
Proof. We fix a δ > 0 and let ν be an arbitrary finite measure on R n of finite 1-energy. Denote F := {z ∈ R n : B(z) ≥ 4A(z)}.
To prove the theorem, it suffices to show that Therefore, Condition (C δ ) is fulfilled.

Theorem 2.3. (i) X satisfies (H) if the following condition holds:
Condition (C log ) : F or any f inite measure ν on R n of f inite 1-energy, there exist a constant ς > 1 and a sequence {y k ↑ ∞} such that y 1 > 1 and Then, for any finite measure ν on R n of finite 1-energy and any ς > 1, there exists a sequence {y k ↑ ∞} such that y 1 > 1 and (2.4) holds.

Proof. Assertion (i):
Let ν be an arbitrary finite measure on R n of finite 1-energy. We choose a constant ς and a sequence {y k } described as in Condition (C log ). We assume without loss of generality that y 1 > ς and (y k ) ς < y k+1 for k ∈ N. Set By (2.1), (2.2) and the dominated convergence theorem, we find that where D 1 > 0 and δ > 0 are constants depending only on ς. The fact that π 2 − arctan x ≤ 1 x for x > 0 has been used above to get the second inequality. Since

Assertion (ii):
Suppose that X satisfies (H). Let ν be an arbitrary finite measure on R n of finite 1-energy and ς > 1 be a constant. By [11,Theorem 5.1], lim λ→∞ c(λ) = 0. We choose an increasing sequence of positive numbers {x k } such that We assume without loss of generality that x k + 1 < x k+1 , k ∈ N. Set To prove (2.4), it suffices to show that For λ > 1, we define f (λ) = log λ. Set By (2.7), we get The proof of Assertion (ii) is complete.
(ii) Suppose X satisfies (H). Then, for any finite measure ν on R n of finite 1-energy and any ς > 1, there exists a sequence of positive numbers {x k } such that N ς The proof of Theorem 2.4 is similar to, but more delicate than that of Theorem 2.3. We put it in the Appendix.
We define Condition (C 0 ) : For any finite measure ν on R n of finite 1-energy, Proof. This is a direct consequence of Theorem 2.4. (C log log ) require only that Im(ψ) is partially well-controlled by 1 + Re(ψ). These weaker conditions are fulfilled by more general Lévy processes and reveal the more essential reason for the validity of (H).

Some consequences and examples
In this section, we will give new examples of Lévy processes satisfying (H). The purpose of these preliminary examples is to exhibit the strength of the criteria obtained in Section 2. Further examples should be expected. In fact, based on Theorems 2.3 and 2.4, there are two ways to further explore Getoor's conjecture. One the one hand, we can construct more general examples of Lévy processes satisfying Condition (C log ), or Condition (C log log ), and hence satisfy (H). One the other hand, we can consider if there exists a (maybe pathological) Lévy process which does not satisfy Condition (C log ) and hence does not satisfy (H).
First, we give the following consequence of Theorem 2.3.
Proof. Suppose Conditions (i) and (ii) hold. Let ν be a finite measure on R n of finite 1-energy. Set Hence Condition (C log ) is fulfilled and the proof is complete by Theorem 2.3(i).
Note that in Condition (ii) of Proposition 3.1 z k+1 may be chosen to be much bigger than z k for each k ∈ N. Hence Proposition 3.1 can be used to construct a large class of Lévy processes with indexes β ′′ < β and satisfying (H). We will give a new class of subordinators satisfying (H) in Example 3.4 below.
To the best of our knowledge, which subordinators satisfy (H) is unknown in general. To appreciate the importance of this problem, we recall below the remarkable result obtained by Glover and Rao. [8]) Let (X t ) t≥0 be a standard Markov process on a locally compact space with a countable base and (T t ) t≥0 be an independent subordinator satisfying Hunt's hypothesis (H). Then (X Tt ) t≥0 satisfies (H).

Proposition 3.3. (Glover and Rao
We choose a sequence {z k } satisfying Let X be a pure-jump subordinator with Lévy measure µ(dx) := ρ(x)dx. Suppose ρ satisfies the following conditions: Then X satisfies (H).
In fact, for z ∈ R with |z| ≥ 1, we have

1)
and Hence Conditions (i) and (ii) of Proposition 3.1 are fulfilled and therefore X satisfies (H).
By Corollary 2.5, we obtain the following proposition. for some constant δ > 0. Then X satisfies (H).
Example 3.6. Let X be a Lévy process on R with Lévy-Khintchine exponent (a, Q, µ). Suppose that there exist constants δ > 0 and c > 0 such that Similar to (3.1), we can show that (3.2) holds. Therefore, X satisfies (H) by Proposition 3.5. Note that in this example it does not matter if a or Q equals 0.
Finally, we give a result on the validity of Hunt's hypothesis (H) for perturbed Lévy processes.
Remark 3.8. In Proposition 3.7, if we assume that X 2 satisfies the Kanda-Forst condition (1.1) or Rao's condition (1.2), then the result is obtained by the corresponding theorems of Kanda-Forst and Rao. But our Proposition 3.7 can be applied to general Lévy processes X 2 satisfying (H), e.g., the X 2 satisfying Conditions (i) and (ii) of Proposition 3.1.

Appendix
Proof of Theorem 2.4.

Assertion (i):
Let ν be an arbitrary finite measure on R n of finite 1-energy. We choose a constant ς and a sequence {x k } described as in Condition (C log log ). We assume without loss of generality that x 1 > 2. We will show below that lim λ→∞ {B(z)≥Nx 1 } λ λ 2 +B 2 (z) |ν(z)| 2 dz = 0. For λ > 1, we define Then, we have where D 1 > 0 and δ > 0 are constants depending only on ς. The fact that π 2 − arctan x ≤ 1 x for x > 0 has been used above to get the second inequality. Since

Assertion (ii):
Suppose X satisfies (H). Let ν be an arbitrary finite measure on R n of finite 1-energy and ς > 1 be a constant. We choose a sequence of increasing natural numbers {p k } satisfying For m = 1, 2 . . ., we set x m,1 = p m + 2 and choose k m such that Define x m,lm = p m +2l m for 1 ≤ l m ≤ k m . We can require without loss of generality that x 1,1 > 1 − 2 log log ς log ς , (4.4) and N ς x 1,1 > e and x m,km < p m+1 , m = 1, 2, . . . Denote F := {z ∈ R n : B(z) ≥ 2A(z)}.
Therefore the proof of Assertion (ii) is complete by noting (2.1).