Hunt's Hypothesis (H) for Pure Jump Lévy Processes and Pure Jump Subordinators

Hunt's hypothesis (H) states that every semipolar set of a Markov process is polar; it is equivalent to the bounded maximum principle, to the coincidence of the fine and cofine topologies up to polar sets, and to continuity of all natural additive functionals. Getoor conjectured some fifty years ago that essentially all Lévy processes satisfy (H), excepting only extremely nonsymmetric cases such as uniform motion, and the conjecture is still open. The sharpest precisely formulated open case is the following question of Hu, Sun and Wang [1]: does every pure jump Lévy process — a process with no Gaussian part, Lévy measure of finite variation, and zero drift coefficient, so that it is exactly the sum of its jumps — satisfy (H)? The critical instance is the subordinator case: a subordinator with nonzero drift provably fails (H), so for subordinators the question is exactly whether every pure jump (zero-drift) subordinator satisfies (H). An affirmative answer in that case would complete the subordination dichotomy: a subordinated process $(X_{\tau_t})$ satisfies (H) if and only if either $(X_t)$ or $(\tau_t)$ does. The problem is completely settled either by a proof for the whole class or by one explicit counterexample.

Background

In probabilistic potential theory one studies which sets a Markov process can hit. The classical framework, laid down by Hunt and codified in the monograph of Blumenthal and Getoor, works with a standard process: a right-continuous, strong Markov process $X=(X_t, P^x)$ on a locally compact state space $E$ with countable base, under the usual hypotheses on its filtration. For a set $D\subset E$ the first hitting time is $T_D=\inf\{t>0:\ X_t\in D\}$. A set $D$ is thin if it is contained in a nearly Borel set $C$ with $P^x(T_C=0)=0$ for every starting point $x$: no point is instantaneously regular for $D$, so the process can enter $D$ only by an actual jump of the trajectory. $D$ is semipolar if it is covered by countably many thin sets; heuristically, the time set $\{t:\ X_t\in D\}$ is at most countable. $D$ is polar if it is contained in a nearly Borel set $C$ with $P^x(T_C<\infty)=0$ for all $x$, meaning the process never reaches it from any starting point. Hunt's hypothesis (H) for $X$ is the statement that every semipolar set of $X$ is polar. It is a basic structural property of the process: when $X$ is in duality with another standard process, (H) is equivalent to the bounded maximum principle for the process potentials, to the fine and cofine topologies differing only by polar sets, and to every natural additive functional (an additive functional carried by a measure) being in fact a continuous additive functional.

A Lévy process is a process with stationary and independent increments on $\mathbb{R}^d$; its law is encoded by the Lévy–Khintchine characteristics $(a, Q, \mu)$: a drift vector, a Gaussian covariance matrix, and a Lévy measure describing the intensity of jumps. A symmetric Lévy process is in duality with its own time reversal and satisfies (H); (H) is also known for processes whose Lévy measure satisfies the Kanda–Forst–Rao condition, a balance condition on the first moments of the small jumps ([2] gives its precise form), and for the further classes produced by the modern line of work of Hu, Sun and coauthors described below. Kesten's classical memoir characterized when single points are hit for one-dimensional processes with independent increments. Nevertheless, some fifty years before [1] Getoor conjectured that essentially all Lévy processes satisfy (H), except for extremely nonsymmetric cases such as uniform motion; this conjecture remains a major unsolved problem of Lévy potential theory. The exception is real: a Lévy process of finite variation with nonzero drift coefficient fails (H), uniform motion being the extreme case.

A subordinator is a one-dimensional increasing Lévy process $\tau$ with drift coefficient $d\ge 0$ and a Lévy measure $\mu$ on $(0,\infty)$ satisfying $\int_0^\infty(1\wedge x)\,\mu(dx)<\infty$; it serves as a random time change. Given a standard process $X$ and an independent subordinator $\tau$, Bochner's subordination forms the standard process $(X_{\tau_t})$. Glover and Rao proved that if $\tau$ satisfies (H), then $(X_{\tau_t})$ satisfies (H). Conversely, Hu and Sun proved that a subordinator satisfying (H) must have zero drift, and Hu, Sun and Wang [1] extend the necessity: any Lévy process with $\int_{\mathbb{R}^d}(1\wedge|x|)\,\mu(dx)<\infty$ that satisfies (H) must have zero drift coefficient. Hence, within subordinators, the only candidates for (H) are the pure jump (zero-drift) ones — and whether all of them satisfy (H) is unknown.

Hu, Sun and Wang [1] call a Lévy process on $\mathbb{R}^d$ pure jump if $Q=0$, $\int_{\mathbb{R}^d}(1\wedge|x|)\,\mu(dx)<\infty$, and the drift coefficient $a'=-(a+\int_{\{|x|<1\}}x\,\mu(dx))$ equals $0$; equivalently $X_t=\sum_{0<s\le t}\Delta X_s$, a process of finite variation on compact time intervals that is exactly the sum of its jumps. (This is stricter than merely having no Gaussian part: for example, a symmetric stable process of index $\alpha\ge 1$ is not pure jump in this sense.) The known positive results in and around this class are substantial but leave the general case open: special subordinators — those whose potential measure $U(A)=E[\int_0^\infty 1_A(\tau_t)\,dt]$ admits a decreasing density on $(0,\infty)$ — satisfy (H) exactly when their drift is zero; locally quasi-stable subordinators satisfy (H); every subordinator of type $(\alpha,\beta)$ (Lévy density pinched between constant multiples of $x^{-1-\alpha}$ and $x^{-1-\beta}$ near the origin) decomposes into the sum of two independent pure jump subordinators each satisfying (H), yet it is unknown whether the type-$(\alpha,\beta)$ process itself does; integral and energy criteria cover many mildly asymmetric Lévy measures; (H) is invariant under locally absolutely continuous changes of measure, which yields new classes of purely discontinuous processes satisfying (H) with asymmetric Lévy densities; (H) for $(X_{\tau_t})$ with a positive-drift subordinator $\tau$ is equivalent to (H) for $X$; deleting an arbitrary finite part of the Lévy measure (adding or removing big jumps) does not affect (H); and compound Poisson processes (finite Lévy measure) satisfy (H). What no available result covers is the strongly asymmetric pure jump process in general. Hu and Sun [2] report having constructed zero-drift subordinators that fail every known sufficient condition and being unable to decide (H) for them — evidence that the question posed at the end of [1] is the genuine frontier of Getoor's conjecture.

Problem Statement

Does every pure jump Lévy process satisfy Hunt's hypothesis (H)?

Here a pure jump Lévy process is a Lévy process on $\mathbb{R}^d$ (any $d\ge 1$) with Lévy–Khintchine characteristics $(a, Q, \mu)$ such that $Q=0$, $\int_{\mathbb{R}^d}(1\wedge|x|)\,\mu(dx)<\infty$, and drift coefficient $a'=-(a+\int_{\{|x|<1\}}x\,\mu(dx))=0$; equivalently, a process of finite variation on compact time intervals that is exactly the sum of its jumps, $X_t=\sum_{0<s\le t}\Delta X_s$. Hunt's hypothesis (H) is the statement that every semipolar set of the process is polar, with thin, semipolar and polar sets understood in the classical sense for standard processes recalled in the Background. The question includes as its critical instance the subordinator case: since a subordinator with nonzero drift coefficient provably fails (H), and a subordinator is automatically pure jump once its drift is zero, the subordinator sub-question is exactly whether every zero-drift subordinator satisfies (H).

A complete answer is required, in either of two forms:

(i) a rigorous proof that (H) holds for every pure jump Lévy process in the class above, in every dimension and for every admissible Lévy measure — which in particular settles the subordinator case; or

(ii) an explicit pure jump Lévy process — concretely, its Lévy–Khintchine data, for example an explicit Lévy measure — together with an explicitly identified set that is semipolar but not polar for that process. A zero-drift subordinator counterexample suffices, since subordinators belong to the stated class, and it refutes the universal statement.

For context (a corollary, not an additional deliverable): an affirmative answer in the subordinator case would complete the subordination dichotomy — for a standard process $(X_t)$ on a locally compact state space with countable base and an independent subordinator $(\tau_t)$, the subordinated process $(X_{\tau_t})$ satisfies (H) if and only if either $(X_t)$ or $(\tau_t)$ satisfies (H) — and would confirm the pure-jump core of Getoor's conjecture.

References

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