Hunt's hypothesis (H) — every semipolar set of a Markov process is polar — is the property that legitimizes classical potential-theoretic reasoning for possibly non-symmetric processes, and Getoor's conjecture that essentially all Lévy processes satisfy (H) has been open for about fifty years. Hu and Sun (2016) proved that a large class of pure-jump subordinators (type-(α,β)(\alpha,\beta)) decompose into sums of two independent subordinators each satisfying (H), and explicitly posed the general closure question: 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)? Their follow-up work resolved only special cases (compound-Poisson summands, condition (S) summands, controlled summands), and their 2021 survey re-poses the question as open (Question 7). The question is open even in dimension one, where a positive answer would settle the companion question whether every one-dimensional Lévy process with ∫(1∧∣x∣) μ(dx)=∞\int(1\wedge|x|)\,\mu(dx)=\infty satisfies (H). The problem asks whether the closure property holds in full generality.

Background

A Lévy process X=(Xt)t≥0X=(X_t)_{t\ge0} on Rn\mathbb{R}^n is a càdlàg (right-continuous with left-hand limits) stochastic process with stationary and independent increments and X0=0X_0=0; its law is encoded by the Lévy–Khintchine exponent ψ\psi with E[ei⟨z,Xt⟩]=e−tψ(z)\mathbb{E}[e^{i\langle z,X_t\rangle}]=e^{-t\psi(z)}, built from a drift vector aa, a symmetric nonnegative-definite matrix QQ (the covariance of the Gaussian part), and a Lévy measure μ\mu describing the jumps.

For a Markov process, a Borel set BB is called polar if Px(TB<∞)=0\mathbb{P}^x(T_B<\infty)=0 for every starting point xx, where TB=inf⁡{t>0:Xt∈B}T_B=\inf\{t>0:X_t\in B\}; BB is called thin if Px(TB=0)=0\mathbb{P}^x(T_B=0)=0 for every xx (the process never enters BB instantaneously), and semipolar if it is a countable union of thin sets. Hunt's hypothesis (H) asserts that every semipolar set is polar. Its importance comes from potential theory: under mild duality conditions, (H) is equivalent to the bounded energy principle, the bounded maximum principle, the bounded positivity principle, and the bounded regularity principle for the potential kernel, so (H) is exactly what makes the classical potential-theoretic calculus valid for a non-symmetric process.

Around 1970, R. K. Getoor conjectured that essentially all Lévy processes satisfy (H), the only exceptions being extremely non-symmetric cases such as deterministic uniform motion; this remains a major open problem of probabilistic potential theory. The known sufficient conditions all constrain the exponent ψ=(a,Q,μ)\psi=(a,Q,\mu) directly: a non-degenerate Gaussian part (QQ of full rank); the Kanda–Forst condition ∣Im ψ(z)∣≤c(1+Re ψ(z))|\mathrm{Im}\,\psi(z)|\le c(1+\mathrm{Re}\,\psi(z)) together with absolute continuity of the resolvent (the time-Laplace transform of the transition semigroup); Rao's integral refinement of it; symmetry of the process; and the solvability condition (S) on the triplet, which requires finitely many jumps outside the range of the Gaussian covariance and the solvability of an explicit linear equation for the drift. Conversely, a subordinator (an increasing Lévy process) that satisfies (H) must be a pure-jump subordinator. In dimension one, the 1971 classification of Bretagnolle, combined with a theorem of Kesten on hitting probabilities of single points, shows that (H) fails exactly in the bounded-variation cases where singletons are semipolar but not polar, and reduces the open part to the case Q=0Q=0 with ∫(1∧∣x∣) μ(dx)=∞\int(1\wedge|x|)\,\mu(dx)=\infty in which no single point is hit.

Hu and Sun (Sci. China Math. 2016) contributed a decomposition theorem: every pure-jump subordinator of type-(α,β)(\alpha,\beta) (Lévy density bounded between constant multiples of x−(1+α)x^{-(1+\alpha)} and x−(1+β)x^{-(1+\beta)} on (0,1](0,1]) is the sum of two independent pure-jump subordinators that each satisfy (H), even though it is still unknown whether the type-(α,β)(\alpha,\beta) subordinator itself satisfies (H). This result led the authors to pose, in the printed text of that paper, the general closure question for sums of independent Lévy processes. Their follow-up paper (Commun. Math. Stat. 2018) settled only special cases: the sum satisfies (H) when one summand is compound Poisson (a pure-jump process with finitely many jumps in bounded time intervals), when both summands satisfy condition (S), or, under existence of resolvent densities, when one exponent is suitably controlled by the other. Their 2021 survey (Acta Math. Sin. Engl. Ser.) re-poses the general question as open (Question 7), records that a positive answer would imply a positive answer to the one-dimensional Question 6 via Kesten's theorem and the decomposition of a process into its positive-jump and negative-jump parts, and lists related invariance results (change of measure; subordination by a positive-drift subordinator) that do not address the spatial-sum question.

Problem Statement

Let X1X_1 and X2X_2 be independent Lévy processes on Rn\mathbb{R}^n, n≥1n\ge1, and suppose that each satisfies Hunt's hypothesis (H), i.e., every semipolar set of XiX_i is polar for i=1,2i=1,2. Does the sum X1+X2X_1+X_2 (the Lévy process with Lévy–Khintchine triplet (a1+a2, Q1+Q2, μ1+μ2)(a_1+a_2,\,Q_1+Q_2,\,\mu_1+\mu_2), or equivalently exponent ψ1+ψ2\psi_1+\psi_2) necessarily satisfy (H)?

The question is open for every dimension n≥1n\ge1, including n=1n=1: either (H) is preserved under the sum of arbitrary independent (H)-satisfying Lévy processes, or there is a pair of Lévy–Khintchine triplets of independent Lévy processes, each satisfying (H), whose sum possesses a semipolar set that is not polar.

References

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