Hunt's hypothesis (H) — every semipolar set of a Markov process is polar — is a cornerstone of probabilistic potential theory, and Getoor's conjecture that essentially all Lévy processes satisfy (H) has been open since the 1960s. Hu and Sun (2018) proved that the sum of two independent Lévy processes satisfies (H) whenever the first summand satisfies (H), resolvent densities exist, and the two Lévy–Khintchine exponents are pointwise comparable, e.g. ∣ψ2(z)∣≤c(1+Re ψ1(z))|\psi_2(z)| \le c(1+\mathrm{Re}\,\psi_1(z)); also when the first summand satisfies (H) and the second is a compound Poisson process; these are sufficient conditions only. The open question (Question 7 of Hu and Sun's 2021 survey) is: if X1X_1 and X2X_2 are independent Lévy processes on Rn\mathbb{R}^n, each satisfying (H), must X1+X2X_1+X_2 satisfy (H)? Settling it requires either a general proof or an explicit counterexample. An affirmative answer would in particular settle the open question of whether every pure-jump subordinator of type-(α,β)(\alpha,\beta) satisfies (H) — each such subordinator is a sum of two independent type-(α,β)(\alpha,\beta) subordinators that do satisfy (H) — and would settle the survey's Question 6, that every one-dimensional Lévy process with ∫(1∧∣x∣) μ(dx)=∞\int(1\wedge|x|)\,\mu(dx)=\infty satisfies (H). The question is listed as open in the 2021 survey.

Background

A Lévy process X=(Xt)t≥0X=(X_t)_{t\ge 0} on Rn\mathbb{R}^n is a stochastic process with stationary and independent increments, started at 00 and continuous in probability. Its law is characterized by the Lévy–Khintchine exponent ψ\psi, defined by E[ei⟨z,Xt⟩]=e−tψ(z)\mathbb{E}[e^{i\langle z,X_t\rangle}]=e^{-t\psi(z)}, where ψ\psi is determined by a triplet (a,Q,μ)(a,Q,\mu): a drift vector aa, the covariance matrix QQ of the Gaussian part, and a Lévy measure μ\mu (a measure on Rn∖{0}\mathbb{R}^n\setminus\{0\} with ∫(1∧∣x∣2) μ(dx)<∞\int(1\wedge|x|^2)\,\mu(dx)<\infty). A set DD is called thin if the process cannot hit DD immediately from any point but can hit it at strictly positive times; a semipolar set is a countable union of thin sets; a polar set is a set that the process, started anywhere, ever hits with probability zero. Hunt's hypothesis (H) asserts that every semipolar set is polar. It is a basic structural principle of probabilistic potential theory: under mild conditions it is equivalent to the bounded energy, maximum, positivity and regularity principles, to the fine and cofine topologies coinciding up to polar sets, and to all natural additive functionals being continuous (see the survey [2]). Around 1966, R. K. Getoor conjectured that essentially all Lévy processes satisfy (H), the only exceptions being extremely nonsymmetric processes such as deterministic uniform motion; this conjecture remains one of the main open problems of the field [2,7].

Sufficient conditions known before the source paper include: symmetry of the process together with existence of resolvent densities (the λ\lambda-resolvent uλ(x)=∫0∞e−λtpt(x) dtu^\lambda(x)=\int_0^\infty e^{-\lambda t}p_t(x)\,dt — the time-discounted transition kernel — having a density with respect to Lebesgue measure); the Kanda–Forst condition ∣Im ψ(z)∣≤c(1+Re ψ(z))|\mathrm{Im}\,\psi(z)|\le c(1+\mathrm{Re}\,\psi(z)) with resolvent densities [8,9]; Rao's integral refinement of it [10]; and a triangle property of the Green function [2]. The only known Lévy-type processes failing (H) are, in dimension one, the bounded-variation processes with nonzero drift that are not compound Poisson — for them singletons are semipolar but not polar, by Bretagnolle's refinement [6] of Kesten's hitting-probability criteria [5] — plus one infinite-dimensional example, Gross's "Bwownian motion" [11]. In dimension one, Bretagnolle's classification (stated as Theorem 2.1 of [1]) leaves essentially two regimes unresolved.

The sum-of-processes viewpoint of Hu and Sun [1,4]: in one of the unresolved one-dimensional regimes, the process XX splits canonically as X=X1+X2X=X_1+X_2 with X1X_1, X2X_2 independent, each spectrally one-sided (all jumps positive, respectively all negative) and of unbounded variation, hence each satisfying (H) by Kesten's theorem [5]. Whether XX itself satisfies (H) is then exactly the question whether (H) is stable under summation. Hu and Sun (2018) [1] prove that (H) is preserved when the second summand is a compound Poisson process (a pure-jump process with finite Lévy measure; Theorem 3.1), when both summands satisfy the geometric condition (S) (Theorem 3.2), and — assuming the summands have resolvent densities — under pointwise comparability of the exponents, e.g. ∣ψ2(z)∣≤c(1+Re ψ1(z))|\psi_2(z)|\le c(1+\mathrm{Re}\,\psi_1(z)) (Theorem 4.1 and Proposition 4.2). These are sufficient conditions only, and nothing is known when the second summand is analytically "larger" than the first. A concrete instance in [1]: a pure-jump subordinator (an increasing Lévy process without Gaussian part or drift) is of type-(α,β)(\alpha,\beta), 0<α<β<10<\alpha<\beta<1, if its Lévy density ρ\rho satisfies c−1x−1−α≤ρ(x)≤c x−1−βc^{-1}x^{-1-\alpha}\le\rho(x)\le c\,x^{-1-\beta} on (0,1](0,1]; every such subordinator is the sum of two independent type-(α,β)(\alpha,\beta) subordinators each satisfying (H) [4], yet it is unknown whether every type-(α,β)(\alpha,\beta) subordinator itself satisfies (H). The general invariance question — is (H) stable under summation? — is Question 7 of the survey [2]; by Remark 5.1 of [2], an affirmative answer would settle the survey's Question 6, namely that every one-dimensional Lévy process with ∫(1∧∣x∣) μ(dx)=∞\int(1\wedge|x|)\,\mu(dx)=\infty satisfies (H).

Problem Statement

Determine whether Hunt's hypothesis (H) is stable under summation of independent Lévy processes. Precise question (Question 7 of [2], already posed in [4] and central to [1]): let X1X_1 and X2X_2 be independent Lévy processes on Rn\mathbb{R}^n, n≥1n\ge 1, each satisfying (H) — that is, every semipolar set of XiX_i is polar. Must X1+X2X_1+X_2 satisfy (H)? A closely related product-process formulation asks whether the Rn+m\mathbb{R}^{n+m}-valued process (X1,X2)(X_1,X_2) must satisfy (H) (Question 4′ of [2], equivalent by the orthogonal-transformation argument of [2] to Question 4 of [2]); by Remark 5.1(ii) of [2], an affirmative answer to that formulation would imply an affirmative answer to the present question. That is, the question is whether the sum satisfies (H) for every dimension nn and every independent pair of Lévy processes each satisfying (H), or whether two independent Lévy processes, each satisfying (H) and with Lévy–Khintchine triplets (ai,Qi,μi)(a_i,Q_i,\mu_i), can be exhibited whose sum admits a semipolar set that is not polar. Two recorded instances fall under this question: (i) does every pure-jump subordinator of type-(α,β)(\alpha,\beta) satisfy (H)? — each is a sum of two independent type-(α,β)(\alpha,\beta) subordinators each satisfying (H) [4], so an affirmative answer settles this instance affirmatively (and a type-(α,β)(\alpha,\beta) counterexample settles the main question negatively); (ii) an affirmative answer settles Question 6 of [2]: every one-dimensional Lévy process with ∫(1∧∣x∣) μ(dx)=∞\int(1\wedge|x|)\,\mu(dx)=\infty satisfies (H).

References

  1. Ze-Chun Hu and Wei Sun, Hunt's Hypothesis (H) for the Sum of Two Independent Lévy Processes, Communications in Mathematics and Statistics 6 (2018) 227–247, DOI 10.1007/s40304-018-0136-y, arXiv:1702.07396, https://doi.org/10.1007/s40304-018-0136-y
  2. Ze-Chun Hu and Wei Sun, Hunt's Hypothesis (H) for Markov Processes: Survey and Beyond, Acta Mathematica Sinica, English Series 37 (2021) 491–508, DOI 10.1007/s10114-021-9343-7, arXiv:1908.06825, https://doi.org/10.1007/s10114-021-9343-7
  3. Ze-Chun Hu, Wei Sun and Li-Fei Wang, Two Theorems on Hunt's Hypothesis (H) for Markov Processes, Potential Analysis 55 (2021) 29–52, DOI 10.1007/s11118-020-09848-2, arXiv:1903.00050, https://doi.org/10.1007/s11118-020-09848-2
  4. Ze-Chun Hu and Wei Sun, Further study on Hunt's hypothesis (H) for Lévy processes, Science China Mathematics 59 (2016) 2205–2226, DOI 10.1007/s11425-015-0600-2, arXiv:1406.2013, https://doi.org/10.1007/s11425-015-0600-2
  5. Harry Kesten, Hitting probabilities of single points for processes with stationary independent increments, Memoirs of the American Mathematical Society, No. 93, American Mathematical Society, Providence, R.I. (1969), DOI 10.1090/memo/0093, https://doi.org/10.1090/memo/0093
  6. Jean Bretagnolle, Résultats de Kesten sur les processus à accroissements indépendants, Séminaire de Probabilités V, Lecture Notes in Mathematics 191, Springer-Verlag, Berlin (1971) 21–36, DOI 10.1007/bfb0058844, https://doi.org/10.1007/bfb0058844
  7. Joseph Glover and Murali Rao, Hunt's Hypothesis (H) and Getoor's Conjecture, The Annals of Probability 14 (1986) 1085–1087, DOI 10.1214/aop/1176992463, https://doi.org/10.1214/aop/1176992463
  8. Mamoru Kanda, Two theorems on capacity for Markov processes with stationary independent increments, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 35 (1976) 159–165, DOI 10.1007/bf00533321, https://doi.org/10.1007/bf00533321
  9. Gunnar Forst, The definition of energy in non-symmetric translation invariant Dirichlet spaces, Mathematische Annalen 216 (1975) 165–172, DOI 10.1007/bf01432544, https://doi.org/10.1007/bf01432544
  10. Murali Rao, Hunt's Hypothesis for Lévy Processes, Proceedings of the American Mathematical Society 104 (1988) 621–624, DOI 10.1090/s0002-9939-1988-0962838-6, https://doi.org/10.1090/s0002-9939-1988-0962838-6
  11. Patrick J. Fitzsimmons, Gross' Brownian motion fails to satisfy the polarity principle, Revue Roumaine de Mathématiques Pures et Appliquées 59 (2014) 87–91, Zbl 1389.60094, https://zbmath.org/1389.60094
  12. Liping Li, Hunt's hypothesis and symmetrization for general 1-dimensional diffusions, arXiv:2107.06163 (2021), https://arxiv.org/abs/2107.06163