Around 1970, R. K. Getoor conjectured that essentially every Lévy process on a finite-dimensional Euclidean space satisfies Hunt's hypothesis (H) — every semipolar set is polar — apart from extremely nonsymmetric degeneracies such as uniform motion. Fifty years of partial results (stable processes, the Kanda–Forst condition and its later relaxations, processes with non-degenerate Gaussian part, special and locally quasi-stable subordinators) have not settled it: the most recent survey (Hu and Sun, 2021) still calls it a major unsolved problem of Markov-process potential theory, and even the pure-jump subordinator case is open. Hansen and Netuka (2016) prove (H) under structural potential-theoretic hypotheses — a translation-invariant balayage space on a locally compact abelian group with a Green function satisfying a local triangle (3G) inequality. The problem is to settle Getoor's conjecture itself: prove that every Lévy process outside the known degenerate classes satisfies (H) — in particular the pure-jump core — or exhibit a non-degenerate Lévy process together with a semipolar set that is not polar.

Background

A Lévy process is a stochastic process (Xt)t≥0(X_t)_{t\ge 0} on Rn\mathbb{R}^n with stationary and independent increments and càdlàg paths; Brownian motion with drift, stable processes, compound Poisson jump processes and subordinators (increasing Lévy processes, used as random time changes) are the standard examples. By the Lévy–Khintchine formula its characteristic function is E[ei⟨z,Xt⟩]=e−tψ(z)\mathbb{E}[e^{i\langle z,X_t\rangle}]=e^{-t\psi(z)}, where the exponent ψ\psi is determined by a drift vector a∈Rna\in\mathbb{R}^n (the drift coefficient of the process, i.e. the coefficient of its deterministic linear component, so that a=0a=0 means the process has no deterministic drift), a symmetric nonnegative-definite matrix QQ (the Gaussian part), and a Lévy measure μ\mu on Rn∖{0}\mathbb{R}^n\setminus\{0\} with ∫(1∧∣x∣2) μ(dx)<∞\int(1\wedge|x|^2)\,\mu(dx)<\infty describing the jumps; a process is called pure jump when a=0a=0 and Q=0Q=0.

Potential theory classifies the exceptional sets of a Markov process by hitting behavior. For a Hunt process (a right-continuous strong Markov process with the usual path regularity), a Borel set BB is polar if, started from any point, the process never hits BB with positive probability; BB is thin if the process does not hit BB instantaneously upon starting from any point of BB, i.e. Px(TB=0)=0P^x(T_B=0)=0 for all xx, where TB=inf⁡{t>0:Xt∈B}T_B=\inf\{t>0:X_t\in B\}; and BB is semipolar if it is covered by countably many thin sets. Semipolar sets are the natural exceptional sets of fine potential theory. Hunt's hypothesis (H) asserts that every semipolar set is polar. Hypothesis (H) is equivalent to several structural facts — the bounded positivity, energy, maximum and regularity principles hold simultaneously, the fine and cofine topologies of a dual pair differ only by polar sets, every natural additive functional is continuous, and Choquet's capacity dichotomy holds (classical results for processes in duality — a situation automatic for Lévy processes on Rn\mathbb{R}^n — reproduced in the Hu–Sun survey) — so (H) is a cornerstone of duality and energy arguments, and its failure disables standard tools.

Getoor conjectured that essentially all Lévy processes satisfy (H), with the exception of extremely nonsymmetric degeneracies such as uniform motion Xt=x+ctX_t=x+ct, whose moving point is a thin but non-polar singleton. The conjecture is known for many classes: compound Poisson processes trivially; all stable processes; more generally processes with resolvent densities whose exponent satisfies the Kanda–Forst bound ∣Im⁡ψ∣≤M(1+Re⁡ψ)|\operatorname{Im}\psi|\le M(1+\operatorname{Re}\psi) and its later relaxations; processes with non-degenerate Gaussian part QQ; and special or locally quasi-stable subordinators without drift. Conversely, (H) provably fails for uniform motion and for subordinators with positive drift (a subordinator satisfying (H) must have drift coefficient 00), and Fitzsimmons showed that an infinite-dimensional Lévy process (Gross's Brownian motion) fails (H), so finite dimensionality is essential.

Hansen and Netuka (2016) contribute a potential-theoretic sufficient condition of a different kind: if the state space is a locally compact abelian group, the cone of excessive functions is translation-invariant and forms a balayage space (an axiomatic potential-theoretic structure in which harmonic functions, potentials and balayage are defined without reference to a particular process), and there exists a Green function G>0G>0 (a kernel through which every potential is represented) with the local triangle (3G) property G(x,z)∧G(y,z)≤C G(x,y)G(x,z)\wedge G(y,z)\le C\,G(x,y) on open neighborhoods, then (H) holds. What remains open is exactly the core of Getoor's conjecture: no known argument removes structural assumptions of this type, and even the pure-jump case is unsettled.

Problem Statement

Let XX be a Lévy process on Rn\mathbb{R}^n (n≥1n\ge 1) with Lévy–Khintchine exponent (a,Q,μ)(a,Q,\mu), and let (H) denote Hunt's hypothesis: every semipolar Borel set is polar. Getoor's conjecture asserts that every 'reasonable' Lévy process satisfies (H) — all except extremely nonsymmetric degeneracies such as uniform motion Xt=x+ctX_t=x+ct. For the purposes of this statement, 'the known degenerate classes' means the Lévy processes on Rn\mathbb{R}^n for which (H) is known to fail, such as uniform motion Xt=x+ctX_t=x+ct and subordinators with positive drift. The degenerate exceptions are partially understood: if XX satisfies (H), Q=0Q=0 and ∫Rn(∣x∣∧1) μ(dx)<∞\int_{\mathbb{R}^n}(|x|\wedge 1)\,\mu(dx)<\infty, then the drift aa must vanish, so that every process with Q=0Q=0, ∫Rn(∣x∣∧1) μ(dx)<∞\int_{\mathbb{R}^n}(|x|\wedge 1)\,\mu(dx)<\infty and non-vanishing drift is likewise known to fail (H); in particular, a subordinator satisfying (H) has drift coefficient 00.

Settle the conjecture, in either direction:

  1. Affirmative resolution. Prove that every Lévy process on Rn\mathbb{R}^n outside the known degenerate classes satisfies (H). This must in particular settle the recognized open core: every pure-jump Lévy process (a=0a=0, Q=0Q=0, μ\mu infinite) — including the one-dimensional case and, most sharply, every pure-jump subordinator (drift coefficient 00, infinite Lévy measure on (0,∞)(0,\infty)) — for which every known sufficient criterion (resolvent densities with ∣Im⁡ψ∣≤M(1+Re⁡ψ)|\operatorname{Im}\psi|\le M(1+\operatorname{Re}\psi), its Rao-type and energy-integral relaxations, non-degenerate Gaussian part, balayage-space structure with a 3G Green function) fails or does not apply.

  2. Negative resolution. Exhibit a specific Lévy process on Rn\mathbb{R}^n, given concretely by its Lévy–Khintchine data (a,Q,μ)(a,Q,\mu) and lying outside the known degenerate classes (such as a pure-jump process), together with an explicit Borel set that is semipolar but not polar, with a rigorous proof of both properties.

Settling the conjecture is equivalent to an exact characterization of the Lévy processes on Rn\mathbb{R}^n satisfying (H) — one that decides the pure-jump core above and is consistent with the known sufficient and necessary conditions. The problem must not be weakened by silently adding structural hypotheses such as symmetry, a non-degenerate Gaussian part, a translation-invariant balayage space with a Green function obeying the local triangle inequality G(x,z)∧G(y,z)≤C G(x,y)G(x,z)\wedge G(y,z)\le C\,G(x,y), or any pointwise control of Im⁡ψ\operatorname{Im}\psi by 1+Re⁡ψ1+\operatorname{Re}\psi.

References

  1. W. Hansen and I. Netuka, Hunt's hypothesis (H) and triangle property of the Green function, Expositiones Mathematicae 34 (1) (2016) 95–100. DOI: 10.1016/j.exmath.2014.12.009, https://doi.org/10.1016/j.exmath.2014.12.009 (preprint arXiv:1411.2900, https://arxiv.org/abs/1411.2900)
  2. Z.-C. Hu and W. Sun, Hunt's Hypothesis (H) for Markov Processes: Survey and Beyond, Acta Mathematica Sinica, English Series 37 (3) (2021) 491–508. DOI: 10.1007/s10114-021-9343-7, https://doi.org/10.1007/s10114-021-9343-7 (preprint arXiv:1908.06825, https://arxiv.org/abs/1908.06825)
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