Does Hunt's hypothesis (H) hold for the lacunary pure-jump subordinators of Hu–Sun–Zhang?

Hunt's hypothesis (H) for a Markov process states that every semipolar set is polar; for Lévy processes it is the subject of Getoor's long-standing conjecture that essentially all Lévy processes satisfy (H). Hu, Sun, and Zhang (Potential Analysis 42, 2015) constructed an explicit family of drift-zero pure-jump subordinators whose Lévy measure has density $x^{-1-\alpha}$ concentrated on a lacunary (super-exponentially spaced) sequence of intervals shrinking to the origin, arranged so that on frequency windows $[n_k/2, 2n_k]$ the imaginary part of the Lévy–Khintchine exponent grows like $n_k^{2\alpha-1}$ while its real part is negligible in comparison, so that $\mathrm{Im}\,\psi(z)/(1+\mathrm{Re}\,\psi(z)) \to \infty$ along the windows. For these subordinators the paper proves that Rao's growth condition fails, and its window estimates likewise rule out the ratio-type Kanda–Forst condition, the paper's own extended Kanda–Forst–Rao condition, and its growth conditions; the authors could not decide whether (H) holds. The problem is to determine, by proof or counterexample, whether these subordinators satisfy (H); no known criterion decides it. The question remained open through the authors' 2021 survey and no later literature resolves it; either answer would be the first decisive information about Getoor's conjecture outside all currently tractable classes.

Background

A Lévy process on $\mathbb{R}^n$ is a stochastic process with stationary independent increments and càdlàg paths (right-continuous with left limits); its law is encoded by the Lévy–Khintchine exponent $\psi$, defined by $\mathbb{E}[e^{i\langle z,X_t\rangle}] = e^{-t\psi(z)}$, where $\psi(z) = i\langle a,z\rangle + \tfrac{1}{2}\langle z,Qz\rangle + \int_{\mathbb{R}^n\setminus\{0\}}(1-e^{i\langle z,x\rangle}+i\langle z,x\rangle\mathbf{1}_{|x|<1})\mu(dx)$. A subordinator is a nondecreasing Lévy process on $[0,\infty)$; it has a nonnegative drift coefficient $d$ and a Lévy measure supported on $(0,\infty)$. In Markov potential theory, a set is polar if the process never hits it (from any starting point, almost surely), and semipolar if it is a countable union of thin sets — sets that the process cannot hit instantaneously from any starting point. Hunt's hypothesis (H) asserts that every semipolar set is polar. The hypothesis is central to the duality theory of Markov processes: for a Lévy process all of whose excessive functions (the nonnegative functions $f$ with $P_t f \le f$ for the transition semigroup $P_t$) are lower semicontinuous — equivalently, by a theorem of Hawkes, a Lévy process with resolvent densities — (H) is equivalent to each of the bounded positivity, energy, maximum, and regularity principles of potential theory, to the statement that the fine and cofine topologies differ only by polar sets, and to a dichotomy of capacity; this lower-semicontinuity hypothesis is carried explicitly by the equivalence theorems (Proposition 1.1 and Theorem 5.1 of the source paper) and is not known to hold for every Lévy process. Getoor conjectured some forty years before the source paper that essentially all Lévy processes satisfy (H) (the conjecture and its name appear in Glover–Rao 1986); it remains the major open problem of the area. The verified partial results are: Kanda (1976) and Forst (1975) proved (H) when the process has bounded continuous transition densities and $|\mathrm{Im}\,\psi(z)| \le M(1+\mathrm{Re}\,\psi(z))$ for a constant $M$; Rao (1988) extended this to the growth condition $|\mathrm{Im}\,\psi(z)| \le (1+\mathrm{Re}\,\psi(z))f(1+\mathrm{Re}\,\psi(z))$ for an increasing $f$ with $\int_N^\infty (\lambda f(\lambda))^{-1}d\lambda = \infty$ for every $N \ge 1$; Hu–Sun (2012) showed that a non-degenerate Gaussian part implies (H) and that a subordinator satisfying (H) must have $d = 0$. For subordinators specifically, (H) is known for special subordinators (whose potential measure has a decreasing density on $(0,\infty)$) with $d=0$, for locally quasi-stable subordinators, and Hu–Sun (2016) proved that any pure-jump subordinator of type-$(\alpha,\beta)$ (Lévy density comparable to $x^{-1-\beta} \lesssim \rho(x) \lesssim x^{-1-\alpha}$ on $(0,1]$) decomposes into two independent subordinators both satisfying (H); Hansen–Netuka (2016) gave an independent sufficient condition, the triangle property of the Green function. The source paper, Hu–Sun–Zhang (2015), added a comparison result showing big jumps do not affect (H), a new necessary and sufficient condition, an extended Kanda–Forst–Rao condition (EKFR, Theorem 4.5), and growth conditions based on lower bounds for $\mathrm{Re}\,\psi(z)/(|z|\log^\gamma|z|)$ (Theorem 4.7, and Proposition 4.10 stated in Remark 4.9). Its Section 5.1 then fixed $\tfrac{1}{2} < \alpha < 1$ and defined, for each $k$, a Lévy measure $\mu_k$ with density $\rho_k(x) = x^{-1-\alpha}$ on the interval $(\tfrac{1}{2n_k^2}, \tfrac{1}{n_k^2})$ and $0$ elsewhere; here $n_1$ is first set to $2$ and then enlarged so that $\tfrac{1}{8}n_1^{2\alpha-1} > \tfrac{6}{1-\alpha}$, and $n_k = (k-1)^{1/(2\alpha-1)}c\,n_{k-1}^{\vartheta}$ for $k \ge 2$ with $\vartheta = \max\{\tfrac{5}{2-2\alpha}, \tfrac{4+2\alpha}{2\alpha-1}\}$ and $c > 0$ depending only on $\alpha$, so the $n_k$ grow super-exponentially and the intervals $(\tfrac{1}{2n_k^2}, \tfrac{1}{n_k^2})$ are pairwise disjoint and shrink to the origin. The associated drift-zero subordinator $X$ has Lévy measure $\mu = \sum_k \mu_k$ and exponent $\psi = \sum_k \psi_k$, where $\psi_k$ is the Lévy–Khintchine exponent of the subordinator with Lévy measure $\mu_k$. On each frequency window $[n_k/2, 2n_k]$ the paper's estimates (5.29)–(5.31) give $\mathrm{Im}\,\psi_k(z) \ge \tfrac{1}{8}n_k^{2\alpha-1}$, $\mathrm{Im}\,\psi(z) \ge \bigl(1-\tfrac{1}{3}\sum_j n_j^{-4}\bigr)\mathrm{Im}\,\psi_k(n_k/2)$, and $\mathrm{Re}\,\psi(z) \le \tfrac{1}{3n_{k-1}^4}\mathrm{Im}\,\psi_k(n_k/2) + 2 + \tfrac{2}{1-\alpha}\sum_j n_j^{-4}$: the exponent is essentially purely imaginary there, so ratio- and growth-type criteria lose traction. The paper proves that Rao's condition fails for these subordinators (estimates (5.32)–(5.34)), and the same window estimates show that the Kanda–Forst ratio condition and the lower-bound conditions of Theorem 4.7 and Proposition 4.10 cannot hold either. The authors state that to the best of their knowledge no existing criterion applies, closing the paper with: 'So far we have not been able to prove or disprove that (H) holds for the subordinators constructed in Subsection 5.1.'

Problem Statement

Determine whether Hunt's hypothesis (H) — every semipolar set is polar — holds for the subordinators constructed in Section 5.1 of Hu, Sun, and Zhang (Potential Analysis 42:585–605, 2015). These are the drift-zero pure-jump subordinators $X$ whose Lévy measure $\mu(dx) = \rho(x)dx$ has, for a fixed $\tfrac{1}{2} < \alpha < 1$, density $\rho(x) = x^{-1-\alpha}$ on the disjoint intervals $(\tfrac{1}{2n_k^2}, \tfrac{1}{n_k^2})$ and $0$ elsewhere, with $n_1$ chosen so that $\tfrac{1}{8}n_1^{2\alpha-1} > \tfrac{6}{1-\alpha}$ and $n_k = (k-1)^{1/(2\alpha-1)}c\,n_{k-1}^{\vartheta}$ for $k \ge 2$, $\vartheta = \max\{\tfrac{5}{2-2\alpha}, \tfrac{4+2\alpha}{2\alpha-1}\}$, $c > 0$ depending only on $\alpha$; writing $\psi_k$ for the Lévy–Khintchine exponent of the subordinator whose Lévy measure has density $x^{-1-\alpha}$ on $(\tfrac{1}{2n_k^2}, \tfrac{1}{n_k^2})$, the exponent of $X$ is $\psi = \sum_k \psi_k$, and on each frequency window $[n_k/2, 2n_k]$ it satisfies $\mathrm{Im}\,\psi_k(z) \ge \tfrac{1}{8}n_k^{2\alpha-1}$, $\mathrm{Im}\,\psi(z) \ge \bigl(1-\tfrac{1}{3}\sum_j n_j^{-4}\bigr)\mathrm{Im}\,\psi_k(n_k/2)$, and $\mathrm{Re}\,\psi(z) \le \tfrac{1}{3n_{k-1}^4}\mathrm{Im}\,\psi_k(n_k/2) + 2 + \tfrac{2}{1-\alpha}\sum_j n_j^{-4}$. Concretely: prove that every semipolar set of $X$ is polar, or disprove (H) by exhibiting a semipolar set of $X$ that is not polar. (The bounded positivity, bounded energy, bounded maximum, and bounded regularity principles are equivalent to (H) for a process whose 1-excessive functions are all lower semicontinuous — equivalently, which has resolvent densities — but this hypothesis has not been verified for the Section 5.1 subordinators, so settling one of those principles decides (H) for $X$ only if the hypothesis is established for $X$ as well.) No existing criterion applies to this construction: the Kanda–Forst–Rao theorem, Rao's growth condition, the extended Kanda–Forst–Rao condition (EKFR), and the growth conditions of Theorem 4.7 and Proposition 4.10 of the source paper all fail for it, while the remaining independent sufficient condition, the Green-function triangle property of Hansen–Netuka, is not known to apply to these subordinators. This is the derived test problem for Getoor's conjecture that essentially all Lévy processes satisfy (H), not the full conjecture itself.

References

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