{"id":"bd080c1a-433d-462c-ac28-54a81b62872c","arxiv_id":"2411.12717","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Excursions of a tree-indexed Markov process away from a recurrent point x form a Poisson point process with intensity given by a new excursion measure, and the genealogy of the excursions is itself a Lévy tree.","lead":"The paper builds a complete excursion theory for Markov processes indexed by random Lévy trees, describing the process between visits to a distinguished point x with a Poissonian structure. A smart generalist should read it because it gives a general tool for random geometry and recovers the Brownian-tree excursion theory of Abraham and Le Gall as a special case.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Poisson excursion theorem inherits its entire local-time indexing from companion [37]; the proof's independent-increments step turns on unverified identities (7.29) and [37, Thm 3.7], so Theorem 7.5 is only as secure as that black box.","rationale":"The reader's CONDITIONAL verdict already identifies the companion paper's local-time construction as the weakest assumption, and my stress-test agrees. The most concrete vulnerable step is in the proof of Theorem 7.5(iii'): independent increments of the excursion process are obtained by showing that the future excursion process E' is the excursion process of an independently constructed P_{0,x,0}-snake. That identification requires (7.29), which is imported from [37], together with the special Markov property [37, Thm 3.7]. If these imported facts fail, the Poisson intensity dt⊗N*_x and the entire excursion representation are unsupported. I saw no internal contradiction in the present manuscript, and the recovery of Abraham-Le Gall's theory in Section 10 is meaningful independent evidence that the framework is coherent. However, the main theorem cannot be fully verified without access to the cited companion statements, so the appropriate verdict remains CONDITIONAL rather than ACCEPT. This does not move the reader's verdict, hence UNCHANGED.","tokens_in":72059,"tokens_out":6652,"duration_ms":70994,"concrete_test":"Obtain the companion paper [37] and verify, line by line, that Theorem 3.7 and equations (4.29)-(4.30) are proved under exactly hypotheses (H1), (H2), (H1_0) and that they imply the support identity supp dA = [0,σ]\\C* and the additivity identity (7.29). As a quantitative analytical check, re-derive (7.29) for a stable tree with ψ(λ)=λ^α, α∈(1,2), and one-dimensional Brownian spatial motion with x=0, computing both sides from the explicit height process; any event of positive P_{0,0,0}-measure on which the two sides differ would invalidate Theorem 7.5.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 7.3 cannot be checked from this manuscript alone. Theorem 7.5 is proved by reducing the Poisson property to (i)-(iii), and (iii) — stationary independent increments — is the load-bearing part. The proof of (iii') cuts the snake at A^{-1}_r, invokes the special Markov property [37, Thm 3.7] and equations (4.29)-(4.30) of [37], and asserts that the future excursion process E' is exactly the excursion process of an independent snake (ρ^1,W^1). The identity (7.29) is the bridge that transfers the A-indexing of excursions after A^{-1}_r to the cumulative exit local time L: if (7.29) is not a formal consequence of [37], then A and L are not on the same time scale and the Poisson intensity dt⊗N*_x is not justified. The same black-box dependence appears in Corollary 7.1, which uses the support identity (7.3), and in Theorem 9.1, which uses [37, Thm 5.1] for the law of the local-time tree. The proof itself flags omitted checks: the recovery of 1_{(0,r]}E from (7.24) is 'left to the reader', and footnote 15 asserts without proof that a coupling argument from [17] adapts to the present path space. No internal contradiction is apparent, and the Abraham-Le Gall consistency check in Section 10 supplies real independent support, but the central claim rises or falls on these imported results.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops an excursion theory for Markov processes indexed by Lévy trees, away from a regular and instantaneous point x of the state space. The main results are the construction of excursion measures N_{x,r} and N*_x, an exit formula (Theorem 7.3), a Poissonian description of the excursion process (Theorem 7.5), and a theorem showing that the genealogy of excursions is encoded in a Lévy tree whose branching masses are the boundary sizes of the excursions (Theorem 9.1). The paper also provides a consistency check with the Abraham–Le Gall excursion theory for Brownian motion indexed by the Brownian tree (Section 10).","tokens_in":72361,"tokens_out":4373,"duration_ms":43894,"significance":"If correct, this is a substantial and original contribution, extending classical excursion theory and Maisonneuve's exit formula to the tree-indexed setting. The Poisson excursion theorem (Theorem 7.5) and the identification of the tree coded by the local time (Theorem 9.1) are natural and potentially powerful tools for Brownian geometry and related models. The paper is openly built on the companion paper [37], but it provides an independent consistency check in Section 10 by recovering the Abraham–Le Gall master formula, which gives real external support. The statements are precise and the overall architecture is coherent.","major_comments":[{"comment":"The proof of property (iii'), stationary and independent increments, is the load-bearing step for the Poisson claim, and it rests on two unproved assertions. The identity (7.29), which transfers the A-indexing of excursions after time A^{-1}_r to the cumulative exit local time L, is stated with only 'a direct comparison' as justification. In addition, the recovery of the restricted excursion measure 1_{(0,r]}E from the pair (7.24) is left to the reader. Since Theorem 7.5 is the central theorem of the paper, these steps should be written out in full or be derived explicitly from the imported statements in [37].","section":"Section 7.3, Theorem 7.5"},{"comment":"The strong Markov property under the infinite measure N_{x,r} is proved by invoking a coupling argument from [17, Lemma 4.1.3], with footnote 15 asserting that the argument adapts to the present path space. This adaptation is not carried out. Proposition 5.4 is used in the proof of the special Markov property under N_{x,r} (Proposition 8.1) and in the derivation of the boundary-size law (Proposition 8.9), so this is a load-bearing technical point that needs a complete proof.","section":"Section 5, Proposition 5.4"},{"comment":"The proof that the processes rX and rX' have identical jumps relies on the claim that the intervals (A_{a_i}, A_{b_i}) are the excursion intervals of rX' above its running infimum, justified only by the support characterization (9.2) and the equality (9.5). These facts are imported from [37, Theorem 5.1] and [37, Theorem 4.19]. While importing results from a companion paper is legitimate, the manuscript should state precisely which properties of the local time A and of the tree T_{H~} are taken from [37] and which are proved here, and it should provide the necessary consequences in a self-contained way. In particular, Lemma 9.2 extends A to N_{x,r} by using Theorem 7.3, but Theorem 7.3 itself already assumes the main properties of A under N_{x,0} from [37].","section":"Section 9, Theorem 9.1"}],"minor_comments":[{"comment":"The display for the decomposition of ρ_t appears to have a missing indicator: the atomic sum should be over 0 < s ≤ t with X_{s-} < I_{s,t}; as printed, the expression is incomplete.","section":"Section 2.2.2, equation (2.16)"},{"comment":"The symbol E is used both for the state space E = E × R_+ (Section 3) and for the excursion point measure E (Section 7.3). This is confusing; a different notation for one of these objects would improve readability.","section":"Throughout"},{"comment":"Since the paper depends so heavily on the companion paper [37], a short table or list summarizing the main imported results (the local time A, the special Markov property, the law of the tree T_{H~}) and indicating where each is used would help the reader and the referees.","section":"Introduction, Section 7"}],"recommendation":"major_revision","confidential_remarks":"The paper is very heavily reliant on the companion paper [37] for the definition and fundamental properties of the local time process A, the special Markov property, and the law of the tree coded by the local time. If [37] is not yet available to the research community, the editor should ensure that it is made available to the referees and that the authors spell out the exact statements from [37] that are used. The external consistency check in Section 10 against the Abraham–Le Gall theory mitigates the risk, but the unproved steps in the proof of Theorem 7.5 and in Proposition 5.4 should be addressed before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nBottom line: this is a serious extension of Itô excursion theory to Markov processes indexed by Lévy trees, and the main Poisson theorem is very likely correct—but the paper is not self-contained, and the proof of that theorem depends on the companion paper [37] in a way that referees will need to check carefully.\n\nWhat's actually new: the excursion measures N_{x,r} and N*_x, the exit formula (Thm 7.3), the Poisson representation for the excursion process (Thm 7.5), and the tree-of-local-time encoding with boundary sizes identified as a Lévy measure (Thm 9.1, Prop 8.9). These go beyond the Brownian/Brownian-tree case of Abraham–Le Gall [2] and beyond the local-time construction in [37]. The paper uses standard machinery—Lévy snakes, spinal decompositions, Poisson calculus—and the architecture is coherent: the exit formula drives the Poisson theorem, and Section 10's consistency check with [2]'s master formula is genuine independent support.\n\nSoft spots, in proportion: the biggest is that the entire indexing of excursions by the local time A is imported from [37], and the stress test is right that Section 7.3 cannot be verified from this manuscript alone. Identity (7.29) and the invocation of [37, Thm 3.7] are load-bearing, and a few details are explicitly left to the reader, including a footnote about a coupling argument adapting from [17]. I don't see an internal contradiction, and the Abraham–Le Gall comparison is real, but a referee must have [37] in hand and should check (7.29) carefully. That's not a fatal flaw—this is a common situation for a two-paper project—but it does mean the present manuscript is not independently checkable.\n\nWho it's for: specialists in Lévy snakes, random trees, and excursion theory. It deserves a serious referee, not a desk reject. The right outcome is likely conditional acceptance with a request to make the dependence on [37] explicit and to supply the omitted checks.\n\nRecommendation: yes, send to peer review. The black-box reliance is legitimate, but the structure and external consistency make it worth the referee time.","headline":"A serious but non-self-contained extension of excursion theory to tree-indexed Markov processes; the main results look right, but referees must verify the reliance on a companion paper.","tokens_in":73004,"tokens_out":3131,"would_cite":true,"duration_ms":31178,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60J80","60J55","60G51","60J25"],"pacs":[],"model":"deepseek-v4-flash","headline":"For a Markov motion on a random Lévy tree, the pieces of path between visits to a point x form a Poisson cloud when marked by a local time.","keywords":["excursion theory","Lévy trees","Lévy snake","local time at a point","Poisson point measure","exit formula","tree coded by the local time","Markov processes indexed by trees"],"falsifier":"Take the Brownian-tree case with linear Brownian spatial motion and x = 0, where the excursion measure is explicit: compute the Laplace functional $\\mathbb{E}[\\exp(-\\int f\\,d\\mathcal{E})]$ for a simple test function f supported on one mark set and one time interval. Theorem 7.5 forces the value $\\exp(-\\int(1-e^{-f})\\,dt\\otimes N^*_x)$; any analytic or simulated deviation from this exponential form would refute the central claim.","tokens_in":71764,"feed_emoji":"🌳","tokens_out":11262,"duration_ms":105209,"temperature":0.7,"pith_summary":"Take a Markov process whose random index is a Lévy tree, a random branching tree built from a spectrally positive Lévy process, and follow the motion away from a regular and instantaneous point x. This paper proves that the tree-indexed process can be decomposed into excursion components away from x, that the excursions marked by a local time at x form a Poisson point measure with explicit intensity, and that collapsing each excursion component to a point produces another Lévy tree whose branching masses are the boundary sizes of the excursions. An exit formula transfers the analysis from the original process to a family of measures $N_{x,r}$ under which the coded tree is still a Lévy tree, which is what makes the theory tractable. If the construction is right, this is the tree-indexed analogue of the classical excursion theorem for time-indexed Markov processes, and it recovers the earlier excursion theory for Brownian motion indexed by the Brownian tree.","feed_headline":"Lévy-tree excursions form a Poisson point measure","feed_subtitle":"A local time at x makes the tree's excursions independent pieces with Lévy-tree genealogy.","key_machinery":"The central object is the Lévy snake $(\\rho,W,\\Lambda)$: a path-valued Markov process whose lifetime process is the height process $H(\\rho)$ of the underlying Lévy tree, so that the tip $W_t(H_t)$ labels the point currently visited by the clockwise exploration. The local time at x, $A$, is defined by (7.1) as the amount of exploration time the labels spend at x, and it is this additive functional that indexes the excursions. The measures $N_{x,r}$ are built by driving the snake with the excursion measure of the underlying Lévy process while the spatial motion follows the exit-system laws of the classical theory; $N^*_x$ is the pruning of $N_{x,0}$ at the first return to x. All arguments are carried out in this $\\mathbb{R}_+$-indexed snake picture, which lets the paper use Poisson calculus, spinal decompositions, and the special Markov property before translating back to the tree.","core_discovery":"The central claim is Theorem 7.5: under the law $P_{0,x,0}$ that starts the Lévy snake at the root with label x and local time 0, the point measure $\\mathcal{E} = \\sum_{u\\in D} \\delta_{(A_{g(u)},\\rho^{u,*},W^{u,*})}$ is a Poisson point measure on $\\mathbb{R}_+ \\times D(\\mathbb{R}_+, M_f(\\mathbb{R}_+) \\times W_E)$ with intensity $dt \\otimes N^*_x$. The excursion measure $N^*_x$ is obtained by truncating the paths of the snake at their first return to x under $N_{x,0}$, and the same setup produces the family $N_{x,r}$ together with the exit formula (7.9), which describes the subtrajectories stemming from the debut points. The paper's second main claim, Theorem 9.1, is that the tree coded by the local time — the tree obtained by identifying each excursion component to a single point — is a $\\tilde{\\psi}$-Lévy tree; its branching points are exactly the excursions with positive boundary size, and the fractal mass of each such point equals that boundary size. In the Brownian-tree case these objects are shown to coincide with the excursion measure and boundary measure of the earlier theory for Brownian motion indexed by the Brownian tree.","pith_inferences":["A natural extension, not developed in the paper, is to use this Poisson structure to prove spatial Markov properties for random surfaces coded by Lévy trees: conditioning on the tree of local time should leave the excursion disks independent with laws given by the boundary-size-conditioned measures.","Because $N^*_x$ is defined by pruning $N_{x,0}$, invariance principles for Lévy snakes should transfer automatically to the excursion measure, giving convergence of discrete approximations to the excursion process without a separate construction of the limit.","The assumption that the label process never hits x at a branching point is load-bearing for unique debuts; models where x is visited at branch points would need a decorated local time and would likely produce a different Poisson intensity.","One direct test of the boundary-size interpretation is to measure excursion boundary sizes in the stable-tree case and compare their law with the Lévy measure $\\tilde{\\pi}$ of the $\\tilde{\\psi}$-tree; Proposition 8.9 predicts an exact match."],"forward_implications":["Excursions away from x, marked by the local time A at their debut, are conditionally independent under $P_{0,x,0}$, with common law $N^*_x$.","The exit formula (7.9) turns calculations about subtrajectories at debuts into integrals against the simpler measures $N_{x,r}$, so the excursion measure inherits the Lévy-tree machinery.","The tree coded by the local time is a $\\tilde{\\psi}$-Lévy tree; excursions with positive boundary size correspond exactly to its branching points, and the boundary size $\\ell_u$ is the fractal mass of the point.","Conditionally on that tree, the positive-boundary excursions are independent with laws $N^{*,\\ell}_x$, while the zero-boundary excursions are independent of the tree and of the positive-boundary family.","In the Brownian-tree special case the theory reproduces the earlier Brownian excursion measure and boundary measure, and the master formula of that theory appears as a particular case of the exit formula."],"supporting_citations":[{"why":"Supplies the local time A at x, its support characterization, the special Markov property, and the fact that the time-changed label process is a \\tilde{\\psi}-Lévy forest; the indexing of excursions depends on it.","marker":"[37]"},{"why":"Provides the Lévy-tree and Lévy-snake framework, the excursion measure of the snake, and the spinal decomposition formulas that the proofs reuse.","marker":"[17]"},{"why":"Defined the earlier theory for Brownian motion indexed by the Brownian tree that the present theory recovers and checks against.","marker":"[2]"},{"why":"Contains the classical exit formula and exit-system setup that Theorem 7.3 is modelled on.","marker":"[33]"},{"why":"Gives the fractal-mass theorem for branching points of Lévy trees used to identify jump sizes of the local-time tree with boundary sizes.","marker":"[18]"},{"why":"Introduces the tree-subordination framework that interprets the tree coded by the local time.","marker":"[27]"}],"fun_headline_variants":["Lévy-tree excursions form Poisson point measure","Excursions of Lévy-indexed Markov processes are Poisson","Local time yields Poisson excursions on Lévy trees","Lévy-tree genealogy of excursion measure","Poisson point measure for Lévy-tree Markov excursions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the companion construction of the local time at x for the Lévy snake, together with the assumption that the label process never hits x at a branching point; if either fails, the excursion components cannot be cleanly defined or indexed.","fun_headline_variants_meta":{"raw":{"variants":["Lévy-tree excursions form Poisson point measure","Excursions of Lévy-indexed Markov processes are Poisson","Local time yields Poisson excursions on Lévy trees","Lévy-tree genealogy of excursion measure","Poisson point measure for Lévy-tree Markov excursions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000218,"raw_usage":{"total_tokens":1433,"prompt_tokens":935,"completion_tokens":498,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":551,"completion_tokens_details":{"reasoning_tokens":425}},"tokens_in":551,"tokens_out":498,"duration_ms":5202,"temperature":1.0,"reasoning_tokens":425,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T17:13:39.060715+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the Brownian-tree case with linear Brownian spatial motion and x = 0, where the excursion measure is explicit: compute the Laplace functional $\\mathbb{E}[\\exp(-\\int f\\,d\\mathcal{E})]$ for a simple test function f supported on one mark set and one time interval. Theorem 7.5 forces the value $\\exp(-\\int(1-e^{-f})\\,dt\\otimes N^*_x)$; any analytic or simulated deviation from this exponential form would refute the central claim.","supporting_citations":[{"cited_title":"Riera, A","cited_arxiv_id":null,"evidence_quote":"Supplies the local time A at x, its support characterization, the special Markov property, and the fact that the time-changed label process is a \\tilde{\\psi}-Lévy forest; the indexing of excursions depends on it."},{"cited_title":"Duquesne, J.-F","cited_arxiv_id":null,"evidence_quote":"Provides the Lévy-tree and Lévy-snake framework, the excursion measure of the snake, and the spinal decomposition formulas that the proofs reuse."},{"cited_title":"Abraham, J.-F","cited_arxiv_id":null,"evidence_quote":"Defined the earlier theory for Brownian motion indexed by the Brownian tree that the present theory recovers and checks against."},{"cited_title":"Maisonneuve , Exit systems","cited_arxiv_id":null,"evidence_quote":"Contains the classical exit formula and exit-system setup that Theorem 7.3 is modelled on."},{"cited_title":"Duquesne, J.-F","cited_arxiv_id":null,"evidence_quote":"Gives the fractal-mass theorem for branching points of Lévy trees used to identify jump sizes of the local-time tree with boundary sizes."},{"cited_title":"Le Gall , Subordination of trees and the Brownian map","cited_arxiv_id":null,"evidence_quote":"Introduces the tree-subordination framework that interprets the tree coded by the local time."}],"review_version":1}