{"id":"2f2080f8-a24f-4a3a-a787-0dec25a38539","arxiv_id":"1908.06825","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For multidimensional Lévy processes with finite jump measure outside the Gaussian range, Hunt's hypothesis is equivalent to the same hypothesis holding for every one-dimensional projection, and new energy decay criteria are proved for products of independent processes.","lead":"This paper surveys Hunt's hypothesis (H) for Markov processes and adds new results for multidimensional Lévy processes, including an equivalence between (H) and (H) for one-dimensional projections under a finiteness condition. Read it for a current map of a 50-year-old open problem in probability and the newest partial answers to it.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.6 depends on Proposition 3.2 from an unpublished preprint, and the projection equivalence is not independently established in this paper.","rationale":"The reader's CONDITIONAL verdict is appropriate. The central new result, Theorem 3.6, gives a clean projection criterion, and the overall proof structure is coherent provided Proposition 3.2 is available and correctly interpreted. Section 4's energy propositions also depend on external results from Kanda [28], but they are more peripheral to the paper's main new claim. I do not see a counterexample to the central claims; the load-bearing weakness is that a key lemma is imported from an unpublished source without proof, and its statement is ambiguous between two notions of drift. A direct verification of the lemma in the exact form needed would settle the matter. The reader's weakest assumption correctly identifies the finiteness condition as the source of the issue, but the more pointed concern is the unproved Proposition 3.2 that converts projection (H) into vanishing drift. Thus the verdict remains CONDITIONAL rather than ACCEPT, and no stronger adjustment is warranted.","tokens_in":19278,"tokens_out":42840,"duration_ms":413760,"concrete_test":"Prove the special case of Proposition 3.2 needed here: if Y_t = c t + C_t, where C is a compound Poisson process and c is nonzero, show directly that a closed interval [a,b] contained in (0,infinity) is thin but not polar, so (H) fails and c must be zero. Then inspect [23, Prop 5.3] and verify that its 'drift coefficient' is the canonical trajectory drift rather than the coefficient a in the (a,0,mu) representation, and that it matches b' = -a - int_{|x|<1} x mu_1(dx) in decomposition (3.4) under this paper's sign convention for the Levy-Khintchine exponent.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In Theorem 3.6, both converses are proved by showing that a one-dimensional projection of X has the form (P b')t + C_t with C compound Poisson, then asserting that Hunt's hypothesis (H) for this projection forces P b' = 0. That assertion is exactly Proposition 3.2, which is cited to the authors' unpublished preprint [23] and is not proved in the present paper. The proposition is not innocent: its phrase 'drift coefficient equals zero' is ambiguous between the coefficient a in the (a,Q,mu) representation and the canonical trajectory drift b = a - int_{|x|<1} x mu(dx). For a pure jump compound Poisson process, a is generally nonzero even though (H) holds, so only the trajectory-drift reading makes Proposition 3.2 true. The paper never fixes this convention, yet the proof of Theorem 3.6 silently uses the trajectory-drift reading to identify b' with the drift of X^(1) in decomposition (3.4). If the convention in [23] differs, or if Proposition 3.2 has an exception for infinite-activity finite-variation jump components, both directions (ii) implies (i) and (iii) implies (i) of Theorem 3.6 fail. The finiteness condition mu(R^n \\ sqrt{Q}R^n) < infinity is what makes the relevant projection processes finite variation, so this gap is exactly as load-bearing as the reader's flagged assumption.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper surveys the state of the art on Hunt's hypothesis (H) for Markov processes and Getoor's conjecture for Lévy processes, and contributes new results on multidimensional Lévy processes from the viewpoints of projections and energy. The main new claims are Theorem 3.6, an equivalence between (H) for a degenerate-Gaussian Lévy process with finitely many jumps outside the range of the Gaussian part and (H) for all of its one-dimensional projections; Theorem 3.7, a related criterion using a subspace strictly between the Gaussian range and the whole space; and several energy results in Section 4, including Propositions 4.5-4.8 on product processes.","tokens_in":19553,"tokens_out":12923,"duration_ms":123991,"significance":"If Theorem 3.6 is correct, it gives a genuinely useful reduction: for a wide class of multidimensional Lévy processes, Hunt's hypothesis can be checked on one-dimensional projections, complementing the classical Kanda-Forst and Rao criteria. The survey portion is well organized and collects many results from the authors' earlier work. The proof of Lemma 3.4 is detailed and correct, and Example 3.5 correctly shows that checking coordinate-axis projections alone is insufficient. However, the central new theorem depends on an unproved and unpublished proposition, and the proof of Proposition 4.8 contains a nontrivial analytical gap; these issues prevent the paper from being accepted in its present form.","major_comments":[{"comment":"Both converses in Theorem 3.6 rely on Proposition 3.2, which is stated in §3.1 without proof and cited to the authors' unpublished preprint [23]. The statement 'its drift coefficient equals zero' is ambiguous: for a process of the form d t plus a compound Poisson process, the coefficient a in the Lévy-Khintchine representation (a,0,µ) includes the small-jump compensation term and is generally nonzero even when the canonical trajectory drift d is zero. The proof of Theorem 3.6 silently uses the canonical-drift reading to conclude b'_i=0 and P2 b'=0. Please include a precise definition of 'drift coefficient' in Proposition 3.2 and either prove the proposition in this paper or cite a published version; as it stands, both (ii)⇒(i) and (iii)⇒(i) of Theorem 3.6 rest on an unverified convention.","section":"§3.2.2, Theorem 3.6"},{"comment":"The proof after equation (4.12) uses a reverse Fatou step to move limsup inside the y-integral. The integrand involves E^{λ+ReΨ(y)}_X(η), but ReΨ(y) ≤ 0 and can be unbounded below, so λ+ReΨ(y) is not positive for all y; the energy E^λ_X is defined only for positive parameter, and the monotonicity of λ ↦ E^λ_X(η) cannot be applied pointwise on the set where λ+ReΨ(y) ≤ 0. A uniform domination argument or a truncation splitting the y-integral is needed. As written, the conclusion lim_{λ→∞} E^λ_Z(µ)=0 is not fully justified.","section":"§4.1, Proposition 4.8"},{"comment":"The final step 'Following the proof of [20, Theorem 1.2, (ii) ⇒ (i)], we can show that F is a polar set of X' is a deferred key argument rather than a proof. In fact, once (X^(1)) is known to satisfy (H) and X^(2) is compound Poisson, the desired conclusion follows directly from Theorem 2.14 (or Proposition 3.9) together with independence of X^(1) and X^(2). Please replace the deferred reference with an explicit citation of Theorem 2.14 and spell out the independence argument.","section":"§3.2.3, Theorem 3.7"}],"minor_comments":[{"comment":"The phrase 'drift coefficient of Y equals zero' inherits the same ambiguity as Proposition 3.2; please use the notation b' from §2.1.2 so that the canonical drift is unambiguous.","section":"§3.1, Proposition 3.3"},{"comment":"The definitions of b and b' appear just before Theorem 2.6 and are used later without being recalled; a short reminder in §3 would improve readability.","section":"§2.1.2, Theorem 2.6"},{"comment":"Reference [23] is listed as an arXiv preprint from 2019; please provide its publication status or a more recent citation if it has appeared.","section":"References"},{"comment":"There are several typographical/OCR issues, e.g., 'Dose any' in Question 6, 'Bwownian' in reference [8], and broken ligatures such as 'exis ting' in the abstract; these should be corrected in the final version.","section":"§5, Question 6 and references"},{"comment":"The sentence 'If ξ charges a semipolar set, then it charges a compact set K ... such that K ⊂ {y ∈ R^m : E^y[exp(-λT_K)] < δ}' would benefit from a brief justification using the definition of semipolar sets as countable unions of thin sets and the regularity of capacity.","section":"§4.1, Proposition 4.6"}],"recommendation":"major_revision","confidential_remarks":"The most important editorial concern is the heavy reliance on the authors' own unpublished preprint [23] for Proposition 3.2, a result that is load-bearing for Theorem 3.6. If the journal does not normally accept proofs that depend on unpublished preprints, this alone would require the authors to include a proof or a published reference. The survey part is useful and the new ideas are promising, but the 'beyond' part needs the identified fixes before it can be certified."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The survey half is solid and worth having: it gives a clear map of Hunt's hypothesis (H) and Getoor's conjecture, with the classical Kanda-Forst-Rao material and the authors' own published contributions organized in a way that is easy to use as a reference. The genuinely new content is the projection material in Section 3 and the energy results in Section 4: Example 3.5 is a clean counterexample to Question 3, Theorem 3.6 gives a plausible projection-based criterion under a finite-measure condition, and Propositions 4.5-4.8 extend Kanda's space-time energy method to products of independent processes. These are real extensions, not repackaged classics.\n\nNow the soft spots, in rough order of severity. Theorem 3.6 depends on Proposition 3.2 from the authors' unpublished preprint [23], and the phrase \"drift coefficient equals zero\" is genuinely ambiguous. In the standard (a,Q,mu) representation, the coefficient a is not the path drift; for a compound Poisson process, a is generally nonzero while (H) holds, so the statement is only true under the trajectory-drift reading. The proof of Theorem 3.6 silently uses that reading to conclude P_i b' = 0, but the paper never fixes the convention. If [23] means something else, or if the intended proposition has exceptions for infinite-activity finite-variation components, both converses in Theorem 3.6 break. This is load-bearing, exactly as the stress-test note says. It is probably fixable, but as written the proof is not self-contained and the cited preprint is not peer-reviewed.\n\nTheorem 3.7 defers a key step to the proof of [20, Theorem 1.2] without adaptation; that is more an inconvenience than a real gap, since [20] is published and the step is standard. Proposition 4.8's use of Fatou for a limsup is sketched: you need a domination argument to justify the swap, and the proof as written is too quick. There is also a minor citation typo: Theorems 2.18 and 2.19 both cite [22, Theorem 2.2], which cannot both be right.\n\nOverall, the central claims look probably correct, but the key new theorem is not yet established as written. I would send this to peer review rather than desk reject, because the survey is valuable and the new ideas deserve scrutiny. The referee should ask for the Proposition 3.2 issue to be resolved, the preprint either published or its proof included, and the energy section's analytic details tightened.","headline":"Useful survey plus a plausible but not fully self-contained new projection criterion; Theorem 3.6 depends on an ambiguous 'drift coefficient' from an unpublished preprint.","tokens_in":20124,"tokens_out":13074,"would_cite":true,"duration_ms":123100,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60G51","60J45","60J25","31C15"],"pacs":[],"model":"deepseek-v4-flash","headline":"For degenerate multidimensional Lévy processes with finite total jump mass outside the Gaussian range, Hunt's hypothesis (H) holds exactly when the projection onto the residual subspace satisfies (H).","keywords":["Hunt's hypothesis (H)","Lévy processes","semipolar sets","polar sets","projections","energy","Getoor's conjecture","potential theory"],"falsifier":"Exhibit a degenerate-$Q$ Lévy process with $\\mu(\\mathbb{R}^n \\setminus \\sqrt{Q}\\,\\mathbb{R}^n)<\\infty$ such that its projection onto $(\\sqrt{Q}\\,\\mathbb{R}^n)^\\perp$ satisfies (H) while the adjusted drift $b'$ does not lie in $\\sqrt{Q}\\,\\mathbb{R}^n$; the theorem predicts that no such process exists, so producing one, or directly showing that $X$ fails (H) while the residual projection satisfies it, would refute the criterion.","tokens_in":19052,"feed_emoji":"📐","tokens_out":16600,"duration_ms":149628,"temperature":0.7,"pith_summary":"This paper surveys what is known about Hunt's hypothesis (H) for Markov processes — the assertion that every semipolar set is polar, i.e. that countable unions of sets essentially never hit at time zero are essentially never hit at all — and adds new criteria for multidimensional Lévy processes. Its central new theorem is a projection criterion: when the Gaussian covariance matrix $Q$ is degenerate and the Lévy measure has finite total mass outside the Gaussian range $\\sqrt{Q}\\,\\mathbb{R}^n$, the process satisfies (H) if and only if its projection onto the orthogonal complement of $\\sqrt{Q}\\,\\mathbb{R}^n$ satisfies (H), which is also equivalent to every one-dimensional projection satisfying (H). The proof uses the Lévy–Itô decomposition to split off a compound Poisson residual and shows that (H) for the residual projection forces the adjusted drift into the Gaussian range, where an earlier equivalence applies. A second set of results shows that finite energy for a product of independent Lévy processes passes to its marginals, and that a singular marginal forces the other marginal to avoid semipolar sets. The survey also records that Getoor's conjecture — that essentially all Lévy processes except the extremely nonsymmetric uniform-motion cases satisfy (H) — remains open, with these results as partial steps.","feed_headline":"One projection decides Hunt's hypothesis for Lévy processes","feed_subtitle":"For degenerate multidimensional Lévy processes, (H) reduces to checking a single projection onto the residual subspace.","key_machinery":"The central machinery is the Lévy–Itô decomposition combined with orthogonal projection. Hunt's hypothesis (H) is the statement that every semipolar set (a countable union of sets that are essentially never hit at time zero) is polar (essentially never hit at any finite time). For the main theorem, the process is written $X_t = X^{(1)}_t + X^{(2)}_t$, where $X^{(1)}$ carries the Gaussian part and the compensated jumps inside $\\sqrt{Q}\\,\\mathbb{R}^n$, and $X^{(2)}$ is a compound Poisson process built from jumps outside that subspace; the finite-mass condition makes $X^{(2)}$ compound Poisson. Projecting onto $(\\sqrt{Q}\\,\\mathbb{R}^n)^\\perp$ removes the Gaussian part, leaving a drift plus compound Poisson process, and validity of (H) for that projection forces the adjusted drift $b'$ into $\\sqrt{Q}\\,\\mathbb{R}^n$. The energy results use the $\\lambda$-energy integrand $\\operatorname{Re}(1/(\\lambda+\\psi(z)))$ and Kanda's comparison estimates to transfer finiteness and vanishing limits between a product process and its marginals; the named Kanda–Forst condition, the inequality $|\\operatorname{Im}\\psi(z)| \\le M(1+\\operatorname{Re}\\psi(z))$, controls the skew part of the Lévy exponent and is used as one factor's hypothesis in the energy theorem.","core_discovery":"The paper's core new result is Theorem 3.6. Let $X$ be an $\\mathbb{R}^n$-valued Lévy process with degenerate Gaussian covariance $Q$, and suppose the Lévy measure $\\mu$ satisfies $\\mu(\\mathbb{R}^n \\setminus \\sqrt{Q}\\,\\mathbb{R}^n)<\\infty$, so that jumps outside the Gaussian range form a compound Poisson process. Then three statements are equivalent: (i) $X$ satisfies Hunt's hypothesis (H); (ii) for every one-dimensional subspace $A$, the projection of $X$ onto $A$ satisfies (H); (iii) the projection of $X$ onto $(\\sqrt{Q}\\,\\mathbb{R}^n)^\\perp$ satisfies (H). The direction (i)$\\Rightarrow$(ii),(iii) is the earlier projection lemma; the new content is that (ii) or even (iii) alone forces (H) for the full process. The proof decomposes $X_t=X^{(1)}_t+X^{(2)}_t$, shows that (H) for the residual projection forces the adjusted drift $b'$ to lie in $\\sqrt{Q}\\,\\mathbb{R}^n$, and invokes the earlier equivalence (Theorem 2.6) between (H) and that drift condition. Theorem 3.7 gives a companion statement with any subspace $S$ satisfying $\\sqrt{Q}\\,\\mathbb{R}^n\\subsetneq S\\subsetneq \\mathbb{R}^n$ and $\\mu(\\mathbb{R}^n\\setminus S)<\\infty$. The energy part adds that for a product $Z_t=(X_t,Y_t)$ of independent Lévy processes with resolvent densities, finite $\\lambda$-energy of a compactly supported measure passes to its marginals; that a finite-energy product measure with a singular factor has the other factor charging no semipolar set; and that if $X$ satisfies (H) and $Y$ satisfies the Kanda–Forst condition, the product measure's $\\lambda$-energy tends to zero as $\\lambda\\to\\infty$.","pith_inferences":["Extension: for concrete models, the projection criterion reduces (H) to a linear-algebra check — whether $b'$ lies in $\\sqrt{Q}\\,\\mathbb{R}^n$ — plus a one-dimensional check on the residual projection, whenever the finite-mass condition holds.","Extension: the coordinate-axis counterexample implies that any automated or experimental test of (H) by projections must scan all directions, or target the orthogonal complement, and cannot rely on coordinate axes alone.","Extension: the energy-transference propositions suggest that verifying Getoor's conjecture could be approached by decomposing a Lévy process into independent summands with complementary energy behavior, if such decompositions can be constructed.","Extension: the boundary of Theorem 3.6 can be probed by building degenerate-$Q$ examples with $\\mu(\\mathbb{R}^n \\setminus \\sqrt{Q}\\,\\mathbb{R}^n)=\\infty$; the paper leaves the projection equivalence open in that case."],"forward_implications":["Under the finite-mass condition, (H) for a degenerate multidimensional Lévy process is equivalent to (H) for its projection onto $(\\sqrt{Q}\\,\\mathbb{R}^n)^\\perp$.","In the same setting, (H) for every one-dimensional projection is equivalent to (H) for the single residual projection, so a single projection suffices.","The coordinate-axis version fails: there is a two-dimensional process whose projections on the two coordinate axes both satisfy (H) while the process itself does not, because the projection on the diagonal $y=-x$ is a uniform motion.","For products of independent Lévy processes, finite $\\lambda$-energy of a product measure forces finite $\\lambda$-energy on each marginal, and a singular marginal forces the other marginal to avoid semipolar sets.","If one factor of a product satisfies (H) and the other satisfies the Kanda–Forst skew-control condition, then the $\\lambda$-energy of any finite-energy product measure vanishes as $\\lambda\\to\\infty$."],"supporting_citations":[{"why":"Provides Theorem 2.6, the equivalence between (H) and the drift condition $b'\\in\\sqrt{Q}\\,\\mathbb{R}^n$ that completes the proof of the projection theorem.","marker":"[20]"},{"why":"Provides Lemma 3.1, that (H) for $X$ passes to every projection, the forward direction of the main theorem.","marker":"[22]"},{"why":"Provides Proposition 3.2 (drift zero for pure-jump finite-variation processes satisfying (H)) and the big-jump removal theorem used in the companion projection theorem.","marker":"[23]"},{"why":"Together with [10], gives the Kanda–Forst theorem, the classical sufficient condition used for the nondegenerate Gaussian component.","marker":"[25]"},{"why":"Companion proof of the Kanda–Forst theorem via energy in translation-invariant Dirichlet spaces.","marker":"[10]"},{"why":"Supplies the energy characterization of (H) and the vanishing-energy criterion for measures charging no semipolar sets, on which Section 4 rests.","marker":"[33]"},{"why":"Supplies the space-time energy lemmas used to transfer energy finiteness and semipolar non-charging between a product process and its marginals.","marker":"[28]"},{"why":"Provides the energy characterization of (H) used in Proposition 4.8, that (H) holds iff all finite 1-energy measures have vanishing $\\lambda$-energy.","marker":"[24]"},{"why":"Establishes the definitions of semipolar and polar sets and the equivalence of (H) with classical potential principles that motivate the survey.","marker":"[2]"}],"fun_headline_variants":["One projection suffices for Hunt's hypothesis in Lévy processes","Single projection decides Hunt's hypothesis for degenerate Lévy processes","Check one projection to verify Hunt's hypothesis for Lévy processes","Hunt's hypothesis for Lévy processes boils down to one projection","One residual projection decides Hunt's hypothesis for Lévy"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the finite-mass condition $\\mu(\\mathbb{R}^n \\setminus \\sqrt{Q}\\,\\mathbb{R}^n)<\\infty$, which ensures that the residual process $X^{(2)}$ is a compound Poisson process; without that condition the projection equivalence of Theorem 3.6 is not established.","fun_headline_variants_meta":{"raw":{"variants":["One projection suffices for Hunt's hypothesis in Lévy processes","Single projection decides Hunt's hypothesis for degenerate Lévy processes","Check one projection to verify Hunt's hypothesis for Lévy processes","Hunt's hypothesis for Lévy processes boils down to one projection","One residual projection decides Hunt's hypothesis for Lévy"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001294,"raw_usage":{"total_tokens":5313,"prompt_tokens":1005,"completion_tokens":4308,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":621,"completion_tokens_details":{"reasoning_tokens":4224}},"tokens_in":621,"tokens_out":4308,"duration_ms":30765,"temperature":1.0,"reasoning_tokens":4224,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:34:18.644046+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a degenerate-$Q$ Lévy process with $\\mu(\\mathbb{R}^n \\setminus \\sqrt{Q}\\,\\mathbb{R}^n)<\\infty$ such that its projection onto $(\\sqrt{Q}\\,\\mathbb{R}^n)^\\perp$ satisfies (H) while the adjusted drift $b'$ does not lie in $\\sqrt{Q}\\,\\mathbb{R}^n$; the theorem predicts that no such process exists, so producing one, or directly showing that $X$ fails (H) while the residual projection satisfies it, would refute the criterion.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides Theorem 2.6, the equivalence between (H) and the drift condition $b'\\in\\sqrt{Q}\\,\\mathbb{R}^n$ that completes the proof of the projection theorem."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides Lemma 3.1, that (H) for $X$ passes to every projection, the forward direction of the main theorem."},{"cited_title":"Two Theorems on Hunt's Hypothesis (H) for Markov Processes","cited_arxiv_id":"1903.00050","evidence_quote":"Provides Proposition 3.2 (drift zero for pure-jump finite-variation processes satisfying (H)) and the big-jump removal theorem used in the companion projection theorem."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Together with [10], gives the Kanda–Forst theorem, the classical sufficient condition used for the nondegenerate Gaussian component."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Companion proof of the Kanda–Forst theorem via energy in translation-invariant Dirichlet spaces."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the energy characterization of (H) and the vanishing-energy criterion for measures charging no semipolar sets, on which Section 4 rests."},{"cited_title":"Nagoya Math","cited_arxiv_id":null,"evidence_quote":"Supplies the space-time energy lemmas used to transfer energy finiteness and semipolar non-charging between a product process and its marginals."},{"cited_title":"Potential Anal","cited_arxiv_id":null,"evidence_quote":"Provides the energy characterization of (H) used in Proposition 4.8, that (H) holds iff all finite 1-energy measures have vanishing $\\lambda$-energy."},{"cited_title":"Academic Press, New York and London (1968)","cited_arxiv_id":null,"evidence_quote":"Establishes the definitions of semipolar and polar sets and the equivalence of (H) with classical potential principles that motivate the survey."}],"review_version":1}