Hunt's hypothesis (H) for a Markov process demands that every semipolar set be polar. For Lévy processes on Rn\mathbb{R}^n it is equivalent to the energy, bounded maximum, and bounded regularity principles, to continuity of all natural additive functionals, and to the dichotomy of capacity, yet it has been verified only for subclasses: processes with non-degenerate Gaussian part, processes satisfying the Kanda–Forst–Rao growth condition on the exponent, special subordinators, and related cases. Getoor conjectured around the mid-1960s that essentially all Lévy processes satisfy (H), the only exceptions being extremely nonsymmetric degenerate cases such as uniform motion, and this conjecture — equivalently, a complete characterization of (H) through the Lévy–Khintchine triplet (a,Q,μ)(a,Q,\mu) — remains open. This record poses the problem at its natural full generality: either prove the conjecture by delimiting the precise exceptional class, or refute it by exhibiting a Lévy process outside the known degenerate classes together with a semipolar set that is not polar.

Background

A Lévy process on Rn\mathbb{R}^n is a stochastic process with stationary and independent increments and paths continuous in probability; Brownian motion with drift, Poisson processes, stable processes (the self-similar Lévy processes, such as the Cauchy process) and subordinators (increasing Lévy processes) are standard examples. By the Lévy–Khintchine formula its law is determined by a triplet (a,Q,μ)(a,Q,\mu), where a∈Rna\in\mathbb{R}^n is a drift vector, QQ is a symmetric nonnegative definite matrix (the Gaussian covariance), and μ\mu is the Lévy measure of jump sizes, through the characteristic exponent ψ(z)=i⟨a,z⟩+12⟨z,Qz⟩+∫Rn(1−ei⟨z,x⟩+i⟨z,x⟩1{∣x∣<1})μ(dx)\psi(z)=i\langle a,z\rangle+\frac{1}{2}\langle z,Qz\rangle+\int_{\mathbb{R}^n}(1-e^{i\langle z,x\rangle}+i\langle z,x\rangle 1_{\{|x|<1\}})\mu(dx), so that E[ei⟨z,Xt⟩]=e−tψ(z)\mathbb{E}[e^{i\langle z,X_t\rangle}]=e^{-t\psi(z)}.

The problem sits in Markov-process potential theory. A Borel set BB is called polar if, when the process is started at any point xx, it almost surely never hits BB; BB is called semipolar if it can be covered by countably many thin sets, a thin set being one that the process does not hit instantly upon starting from any of its points. Intuitively, polar means "never hit" and semipolar means "hit at most countably many times". Hunt's hypothesis (H) for the process asserts that every semipolar set is polar. Its importance comes from a chain of equivalences established by Blumenthal and Getoor, Rao, and Fitzsimmons: under lower semicontinuity of the excessive functions (the nonnegative functions ff with Ptf≤fP_t f\le f for every tt under the transition semigroup (Pt)(P_t), i.e. the superharmonic functions of the process), (H) is equivalent to the energy, bounded maximum, and bounded regularity principles, three classical principles for the potentials of finite measures under the resolvent of the process; independently of that, (H) holds if and only if every natural additive functional (a path functional that accumulates a measure along the trajectory without anticipating the future) is continuous, if and only if the fine and cofine topologies (the coarsest topologies making all excessive, respectively co-excessive — excessive for the time-reversed process — functions continuous) differ only by polar sets, and if and only if the capacity dichotomy of Fitzsimmons and Kanda holds, i.e. every compact set contains two disjoint subsets of the same capacity. For symmetric processes the Dirichlet-form theory of Silverstein (a Dirichlet form is the closed symmetric bilinear form associated with the transition semigroup) already yields that semipolar sets are polar outside a set null for the symmetrizing measure, so (H) is genuinely a question about nonsymmetric processes.

Considerable partial results exist. The Kanda–Forst theorem states that if XX has bounded continuous transition densities and ∣Im ψ∣≤M(1+Re ψ)|\mathrm{Im}\,\psi|\le M(1+\mathrm{Re}\,\psi) for some constant MM, then XX satisfies (H); Rao extended this by allowing ∣1+ψ∣|1+\psi| to grow like (1+Re ψ) f(1+Re ψ)(1+\mathrm{Re}\,\psi)\,f(1+\mathrm{Re}\,\psi) for a suitable increasing function ff of at most logarithmic growth, which yields that all stable processes on Rn\mathbb{R}^n satisfy (H). Compound Poisson processes (pure-jump processes with only finitely many jumps in any bounded time interval) satisfy (H) trivially. Hu and Sun proved that if the Gaussian matrix QQ is non-degenerate then XX satisfies (H), and that under the finiteness condition μ(Rn∖QRn)<∞\mu(\mathbb{R}^n\setminus\sqrt{Q}\mathbb{R}^n)<\infty the validity of (H) is equivalent to the solvability of the linear equation Qy=−a−∫{x∈Rn∖QRn: ∣x∣<1}xμ(dx)\sqrt{Q}y=-a-\int_{\{x\in\mathbb{R}^n\setminus\sqrt{Q}\mathbb{R}^n:\,|x|<1\}}x\mu(dx) in y∈Rny\in\mathbb{R}^n (condition (S)). On the negative side, a subordinator satisfying (H) must have zero drift coefficient, and more generally a pure-jump process of finite variation (zero Gaussian part and ∫Rn(∣x∣∧1) μ(dx)<∞\int_{\mathbb{R}^n}(|x|\wedge 1)\,\mu(dx)<\infty) satisfying (H) must have zero drift; deterministic uniform motion Xt=btX_t=bt with b≠0b\neq 0 fails (H). Beyond these, Hu, Sun and coauthors developed energy-based necessary and sufficient conditions, an extended Kanda–Forst–Rao condition, verified (H) for special subordinators (those whose potential measure has a decreasing density) and locally quasi-stable subordinators, proved that big jumps have no effect on (H), and proved invariance of (H) under absolutely continuous measure changes, under subordination with positive drift, and under projections. Nevertheless, no characterization of (H) through the triplet is known: the conjecture recorded by Glover and Rao — stated by Getoor some fifty years before the 2019 survey of Hu and Sun, which calls it a major unsolved problem in the potential theory for Markov processes — is that essentially all Lévy processes satisfy (H), except for some extremely nonsymmetric cases like uniform motions, and it remains open.

Problem Statement

Decide Getoor's conjecture at its natural, unrestricted generality for Lévy processes on Rn\mathbb{R}^n: determine exactly which Lévy–Khintchine triplets (a,Q,μ)(a,Q,\mu) give rise to Lévy processes satisfying Hunt's hypothesis (H), i.e. processes for which every semipolar set is polar. Concretely, the two directions of resolution are the following.

(1) Prove the conjecture: show that every Lévy process on Rn\mathbb{R}^n satisfies (H) outside a precisely delimited exceptional class of extremely nonsymmetric degenerate processes containing uniform motion, and identify that class explicitly in terms of the triplet. Such a proof is necessarily consistent with the known landmarks: processes with non-degenerate Gaussian part satisfy (H); under μ(Rn∖QRn)<∞\mu(\mathbb{R}^n\setminus\sqrt{Q}\mathbb{R}^n)<\infty validity of (H) is equivalent to the solvability of Qy=−a−∫{x∈Rn∖QRn: ∣x∣<1}xμ(dx)\sqrt{Q}y=-a-\int_{\{x\in\mathbb{R}^n\setminus\sqrt{Q}\mathbb{R}^n:\,|x|<1\}}x\mu(dx); a subordinator, and more generally a pure-jump process of finite variation, satisfies (H) only if its drift coefficient vanishes; every stable process and every compound Poisson process satisfies (H). In particular this would settle Questions 5, 6 and the positive direction of Question 7 posed in the survey of Hu and Sun: whether every pure jump subordinator with infinite Lévy measure — more generally, every pure jump Lévy process outside the known degenerate classes — satisfies (H); whether every one-dimensional Lévy process with ∫R(1∧∣x∣) μ(dx)=∞\int_{\mathbb{R}}(1\wedge|x|)\,\mu(dx)=\infty satisfies (H); and whether a sum of two independent Lévy processes satisfying (H) again satisfies (H).

(2) Refute the conjecture: exhibit a concrete Lévy process, specified by its triplet, that lies outside the known degenerate failure classes (deterministic uniform motion, and nonzero drift in the pure-jump finite-variation directions excluded by the necessity results above), together with an explicit semipolar set for this process that is not polar.

The informal phrase "essentially all, except extremely nonsymmetric cases" is part of the source problem and must be made precise by the answer, not circumvented: answers restricted to a subclass (for example, only subordinators, only one-dimensional processes, or only processes satisfying a growth condition on ψ\psi) do not resolve the problem as stated.

References

  1. Ze-Chun Hu, Wei Sun, "Hunt's Hypothesis (H) for Markov Processes: Survey and Beyond", arXiv:1908.06825; published in Acta Mathematica Sinica, English Series 37(3):491–508 (2021), DOI 10.1007/s10114-021-9343-7, https://arxiv.org/abs/1908.06825
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