Pith. sign in

REVIEW 3 major objections 5 minor 1 cited by

General diffusions on metric graphs as limits of time-space Markov Chains

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper proves that a space-time Markov chain on a subdivision of a metric graph approximates any general diffusion on the graph in p-Wasserstein distance with rate $|\Delta|_X^\alpha$ for every $\alpha < \tfrac14 \wedge \tfrac1p$, and…

desk verdict Genuine extension with useful explicit transition formulas, but the central rate theorem rests on an unjustified conditional-expectation swap in Proposition 4.1. read the letter →

arxiv 2507.23724 v1 pith:KSA6QLEY submitted 2025-07-31 math.PR

Alexis Anagnostakis This is my paper · ORCID
classification math.PR MSC 60J6060J5535J0860J50
keywords diffusiononnetworkMarkovchainapproximationrandomwalkinvarianceprinciplegeneralizedsecondorderdifferentialoperatorspathregularitymetricgraphWassersteindistance
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper introduces a random-walk approximation, the Space-Time Markov Chain Approximation (STMCA), for a general diffusion on a finite metric graph, and proves that the approximation converges to the diffusion in p-Wasserstein distance at a controlled rate. The rate is expressed through a thinness quantifier of the subdivision: for any exponent $\alpha$ below $\tfrac14 \wedge \tfrac1p$, the distance is bounded by a constant times $|\Delta|_X^\alpha$, and with adapted subdivisions the same mechanism doubles the rate in the maximal cell size. The proof couples the diffusion and the chain through the embedded skeleton of the diffusion, so that the chain's positions and jump times match the diffusion's skeleton in distribution. The paper also gives explicit formulas for the transition probabilities and conditional transition times, so the scheme can be run numerically.

What carries the argument

The Space-Time Markov Chain Approximation (STMCA) is a $V_\Delta$-valued random walk whose transitions are $(p_{x,y}, t_{x,y})$, where $p_{x,y}=P_x(T_{U_x}=T_y)$ is the probability that the diffusion exits the cell centered at $x$ through $y$, and $t_{x,y}=E_x(T_{U_x}|T_{U_x}=T_y)$ is the conditional expected exit time. The embedding property of Proposition 2.13 identifies the STMCA path with the diffusion sampled at the random times $\tau^\Delta_{K(t)}$, reducing Wasserstein distance to bounds on $|\tau_{K(t)}-t|$ and on the path regularity of $X$. The thinness quantifier $|\Delta|_X$, defined via re-oriented scale and speed measures on vertex neighborhoods and edge segments, controls the time-change error; explicit Green-function and Dirichlet-problem formulas (Propositions 3.1–3.6) supply the transition data. Moment bounds for the embedding times (Proposition 4.1) and Kolmogorov-type regularity estimates (Section 5) are the remaining ingredients.

What would settle it

Check on a two-edged star graph with asymmetric spinning probabilities and positive stickiness $\rho$ whether the two quantities $E_x[\tau_k-\tau_{k-1}|\mathcal{F}_{\tau_{k-1}}]$ and $E_x[\tau_k-\tau_{k-1}|X_{\tau_k},X_{\tau_{k-1}}]$ have equal cumulative sums along a skeleton path; if their sums differ by more than the allowed $O(|\Delta|_X)$ error, Proposition 4.1 fails and Theorem 2.10's main estimate does not follow.

Watch

Extended reading notes

Core claim

Theorem 2.10 states that for an NSE (natural-scale-on-edges) general diffusion on a finite metric graph, the STMCA on any covering subdivision $\Delta$ satisfies $W_p^T(X,\tilde X^\Delta) \le C |\Delta|_X^\alpha$ for every $\alpha \in (0, \tfrac14 \wedge \tfrac1p)$ and $\varepsilon>0$, uniformly over $(\varepsilon,V)$-symmetric covering subdivisions, provided the diffusion satisfies a H\"older-regularity condition (Condition 2.8) or a speed-measure lower bound (Condition 2.9). This is claimed to be the first explicit quantitative convergence rate for random-walk approximations of general diffusions on metric graphs, and it implies convergence in law as the thinness quantifier goes to zero. With subdivisions adapted so that $|\Delta|_X \lesssim |\Delta|^2$, the bound becomes $O(|\Delta|^{2\alpha})$, i.e. a rate arbitrarily close to $\tfrac12 \wedge \tfrac2p$ in the maximal cell size. The central identity making the result possible is the embedding property: if $\tau_k$ are the successive hitting times of the subdivision vertices and $K(t)$ is the random counter built from skeleton-conditional expected jump times, then $X_{\tau_{K(t)}}$ has the same law as the STMCA at time $t$.

Load-bearing premise

The proof of the key embedding-time bound assumes that conditioning on the full past before a jump gives the same expected jump length as conditioning only on the two vertices the jump connects; this equality is asserted without proof, and the bound on the time error collapses if it fails.

Editorial extensions

If this is right

  • For every $T>0$, the STMCA processes converge in law to the diffusion as the thinness quantifier $|\Delta|_X$ tends to zero (Corollary 2.11).
  • The explicit formulas for transition probabilities and times make the scheme implementable, with asymptotics near vertices that simplify the sticky case.
  • Adapting the subdivision so that $|\Delta|_X \le |\Delta|^2$ pushes the convergence rate in the maximal cell size up to any exponent below $\tfrac12 \wedge \tfrac2p$.
  • The construction generalizes the one-dimensional STMCA, the classical invariance principle, sticky random walks, and oscillating random walks to general metric-graph diffusions.
  • The Wasserstein bound is uniform over $(\varepsilon,V)$-symmetric covering subdivisions, so the rate statement holds for a whole family of discretizations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: If the embedding-time identification holds, the same skeleton-coupling argument should yield explicit constants in the rate, making the STMCA usable as a certified Monte Carlo method for diffusions on networks.
  • Editorial inference: The thinness quantifier suggests a natural adaptive mesh criterion: refine cells where the speed measure is large or where boundaries are sticky, which the numerics indicate matters for reproducing boundary repulsion.
  • Editorial inference: The restriction to NSE diffusions may be largely removable; the author notes that many non-NSE cases reduce to NSE by a transformation, so the scheme likely extends to skew diffusions on graphs after a deterministic change of coordinates.
  • Editorial inference: A direct testable extension is to replace the p-Wasserstein bound with a path-space strong approximation result, which would follow if the embedding times $\tau_{K(t)}$ could be shown to be close to $t$ in a stronger sense than $L^2$.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper introduces the Space-Time Markov Chain Approximation (STMCA) for general diffusions on finite metric graphs: a random walk on subdivisions whose transition probabilities and conditional holding times match those of the target diffusion. The main result (Theorem 2.10) asserts a quantitative Wasserstein convergence rate O(|Delta|_X^alpha) for alpha < 1/4 wedge 1/p under a regularity condition, with improved rates for adapted subdivisions. The proof proceeds by an embedding property (Proposition 2.13), embedding-time estimates (Proposition 4.1), regularity estimates (Section 5), and a final Wasserstein estimate (Section 6). The paper also gives explicit Green-function formulas for the transition quantities and numerical experiments on star graphs.

Significance. If Theorem 2.10 holds as stated, it is a meaningful contribution: it appears to give the first explicit convergence rate for random-walk approximations of general diffusions on metric graphs, and the explicit transition formulas make the scheme implementable. The embedding construction and the extension of the one-dimensional STMCA framework to metric graphs are natural and potentially useful. The numerical experiments illustrate sticky and boundary effects, although they do not verify the rate. The main theorem is not a definitional circularity: it extends earlier work and contains no fitted parameters. However, the written proof of the central embedding-time estimate has a genuine gap, and the regularity section relies heavily on the author's unpublished preprint [2]; both need to be addressed.

major comments (3)
  1. [Section 4.1, Eqs. (4.1)-(4.2)] The proof of Proposition 4.1 conflates two different conditional centerings. Equation (4.1) writes the error as tau_{K(t)} - sum_{k=1}^{K(t)} E[tau_k - tau_{k-1} | F_{tau_{k-1}}], but the martingale M_n of Lemma 4.2 is centered at E[tau_k - tau_{k-1} | X_{tau_k}, X_{tau_{k-1}}]. The equality in (4.2) between the L^2 norm of the F_{tau_{k-1}}-centered sum and the sum of skeleton conditional variances is therefore not justified; in fact the two centered sums differ by sum_k (E[cdot | F_{tau_{k-1}}] - E[cdot | skeleton]), which is not a martingale increment and is not controlled anywhere. Since Proposition 4.1 supplies the quantitative embedding-time bound used in Theorem 2.10, this is a load-bearing gap. The proof can likely be repaired by rewriting (4.1) with the skeleton-conditional sums throughout and bounding the second term by the maximal one-step skeleton holding time, but as written the central estimate does not follow.
  2. [Section 4.1, Eq. (4.3) and Section 2.5] The second term in (4.1) is bounded in (4.3) by sup_{y,j} v_j^1(y)/v_j^0(y) <= K_Delta |Delta|_X, but K(t) is defined in Section 2.5 using cumulative sums of E[tau_k - tau_{k-1} | X_{tau_k}, X_{tau_{k-1}}], not of the F_{tau_{k-1}}-conditional expectations appearing in (4.3). The difference between the two cumulative sums is a sum of K(t) terms, and K(t) is typically of order T/|Delta|_X, so the accumulated discrepancy is not bounded by the one-step maximum. Without an additional estimate comparing the two sums, the bound on |t - sum E[tau | F]| does not follow from the definition of K(t).
  3. [Section 5, Proposition 5.5 and Lemma 5.6] The regularity estimates that feed Theorem 2.10 depend on the time-change representation in [2, Theorem 2.14] and on [6, Theorem 3.1], and the proof of Proposition 5.5 contains several substantial steps that are only indicated, such as 'by excursion flipping', 'by the time-change characterization', and the claim that (5.2) is immediate. In particular, the construction of the process Z with speed measure m_Z and the local-time change-of-variables identities following (5.3) would need to be written out, and the stochastic domination in Lemma 5.6, P_v(T^v_{l*} < T) <= P_v(T^Z_{l*} < T), is asserted rather than proved. Since Condition 2.9 implying Condition 2.8 is one of the two sufficient conditions for Theorem 2.10, this chain of unstated dependencies makes the main theorem conditional on [2].
minor comments (5)
  1. [Section 2.2] The notation |U|_X is defined separately for vertex neighborhoods and edge-segments; consider unifying the two cases in one displayed definition to avoid ambiguity.
  2. [Theorem 2.10] The statement says 'for every x >= 0', but x is an element of the metric graph Gamma; it should read 'for every x in Gamma'.
  3. [Throughout] There are numerous typos and small grammatical errors, including 'elge-lengths' (Section 2.1), 'poiting' (Section 2.1), 'spate space' (Proposition 2.13), 'vells' (proof of Proposition 3.7), 'Conditon' (Section 5.1), and 'the the maximum' in the abstract.
  4. [Section 2.4] The definition of W_p^T has a formatting issue: the supremum and the L^p norm should be displayed unambiguously so that the metric on path space is clear.
  5. [Proposition 4.5] The proof references 'Lemmata C.1 and 4.4' but the displayed bound after conditioning on B applies Lemma 4.4 to each increment; please clarify the exact role of Lemma C.1 in that step.

Circularity Check

1 steps flagged · score 2.0 of 10

No definitional circularity: the STMCA convergence theorem is a genuine extension, but the proof relies on a load-bearing self-citation and has a filtration-mixing gap in Proposition 4.1 that is a correctness risk rather than a circular reduction.

  1. self citation load bearing [Section 5.1, proof of Proposition 5.5; announced in Section 1]
    "By [2, Theorem 2.14], there exists a Walsh Brownian motion W = (J,R) with the same bias parameters (βe; e∈E) as X, defined on an extension of Px such that X = (Wγ(t))t≥0, where t↦γ(t) is the right-inverse of A(t)."

    The regularity estimate (Proposition 5.2) needed for Theorem 2.10's Condition 2.9 branch is proved by invoking this time-change representation from the author's own unpublished preprint [2]. This is load-bearing: without it, the moment bounds for star-graph diffusions would not follow as written. However, [2] is a separate characterization theorem about star-graph diffusions, not a statement of STMCA convergence, and it is parameter-free with stated assumptions that do not include the target result. It is therefore independent evidence rather than a definitional circle around the paper's central convergence claim, though it does make the Condition 2.9 branch depend on an unrefereed self-citation.

full rationale

I find no step where a predicted quantity is a fitted input or where a derived result is its own definition by construction. The STMCA is genuinely constructed from the diffusion's hitting probabilities and conditional exit times, and Theorem 2.10 establishes a quantitative Wasserstein bound through a coupling and nontrivial estimates on embedding times and path regularity. Proposition 2.13 is essentially a coupling built into the definitions of the STMCA and the counter K(t), but it is used as a coupling device, not as an independent convergence prediction. The main issues are two. First, a load-bearing self-citation: the proof of Proposition 5.5 uses [2, Theorem 2.14] from the author's own preprint; this is real evidence of a separate theorem, so it does not make the argument circular, but it means the Condition 2.9 branch is not self-contained. Second, a correctness gap, not a circularity: in the proof of Proposition 4.1, equations (4.1)-(4.2) replace E[τ_k−τ_{k−1} | F_{τ_{k−1}}]-centered sums by skeleton-conditional variances Var[τ_k−τ_{k−1} | X_{τ_k}, X_{τ_{k−1}}] without proving the required identification; the equality appears to mix the natural filtration at the stopping time with the skeleton sigma-algebra. If this identification fails, the L2 embedding-time bound in Proposition 4.1 is not justified as written, and Theorem 2.10 loses its main estimate. That is a proof gap affecting correctness, not a self-definitional or fitted-input circularity. No parameters are fitted, no external benchmark is renamed, and the paper does not import a uniqueness theorem to forbid alternatives. Accordingly, the circularity score is low.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the class of general diffusions defined by A1-A2, on the NSE restriction, on a regularity condition (2.8 or 2.9), and on symmetry of subdivisions. The proof additionally assumes the time-change characterization from the author's preprint [2] and the 1D regularity theorem from [6]. No parameters are fitted to data. No new physical entities are introduced.

assumptions (6)
  • domain assumption A1: edge generator is L_e = (1/2) D_{m_e} D_{s_e} with scale and speed, excluding killing behavior.
    Defines the class of diffusions considered; used throughout the paper from Section 2.1 onward.
  • domain assumption A2: lateral Wentzell-type conditions at vertices: sum_j beta_j f'(j,0) = rho_v L_e f(e,0).
    Defines spinning and sticky behavior at vertices; used in the transition probability formulas and in Proposition 3.3.
  • domain assumption NSE condition: the scale function on every edge has derivative identically +1 or -1.
    Theorem 2.10 and Proposition 3.7 assume natural scale on edges; non-NSE cases are only addressed in a remark.
  • domain assumption Condition 2.8 or Condition 2.9: Lp Holder-1/2 regularity in time, or a lower bound on speed measures.
    Needed for the regularity estimates in Section 5; if neither condition holds, the proof of Theorem 2.10 does not go through.
  • domain assumption Subdivisions are covering and (epsilon,V)-symmetric.
    Used to make the constant K_Delta depend only on epsilon; without symmetry the bounds in Proposition 3.7 blow up.
  • ad hoc to paper The time-change representation of general star-graph diffusions as time-changed Walsh Brownian motion (Theorem 2.14 of [2]).
    Load-bearing for Proposition 5.5 and hence for Section 5; this is the author's own unpublished preprint and is not proved in this paper.

how reviews work

0 comments
Cite this review

Pith. "Pith review of General diffusions on metric graphs as limits of time-space Markov Chains." pith.science (2026). https://pith.science/paper/KSA6QLEY

@misc{pith2026250723724,
  author       = {Pith},
  title        = {Pith review of: General diffusions on metric graphs as limits of time-space Markov Chains},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KSA6QLEY}},
  note         = {Machine review of arXiv:2507.23724}
}
abstract

We introduce the Space-Time Markov Chain Approximation (STMCA) for a general diffusion process on a finite metric graph $\Gamma$. The STMCA is a doubly asymmetric (in both time and space) random walk defined on a subdivisions of $\Gamma$, with transition probabilities and conditional transition times that match, in expectation, those of the target diffusion. We derive bounds on the $p$-Wasserstein distances between the diffusion and its STMCA in terms of a thinness quantifier of the subdivision. This bound shows that convergence occurs at any rate inferior to $\frac{1}{4} \wedge \frac{1}{p} $ in terms of the the maximum cell size of the subdivision, for adapted subdivisions, at any rate inferior to $\frac{1}{2} \wedge \frac{2}{p} $. Additionally, we provide explicit analytical formulas for transition probabilities and times, enabling practical implementation of the STMCA. Numerical experiments illustrate our results.

Figures

Figures reproduced from arXiv: 2507.23724 by the authors.

Figure 1
Figure 1. Sample path of the STMCA for the diffusion from Example 7.1, of initial value [PITH_FULL_IMAGE:figures/full_fig_p029_1.png] view at source ↗
Figure 2
Figure 2. Kernel density approximation (kde) of the STMCA for the diffusion from [PITH_FULL_IMAGE:figures/full_fig_p030_2.png] view at source ↗
Figure 3
Figure 3. Histograms, kernel density estimations, and number of samples on every edge [PITH_FULL_IMAGE:figures/full_fig_p031_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Markov Chain Approximation of Sticky Diffusions and Hamilton-Jacobi-Bellman Equations on Networks

    math.NA 2026-07 accept novelty 6.0 of 10

    An edge-adapted Markov chain with probabilistic vertex residence converges weakly to sticky/Kirchhoff network diffusions and yields a convergent fully discrete semi-Lagrangian HJB scheme.

Reference graph

Works this paper leans on

42 extracted references · 40 canonical work pages · cited by 1 Pith paper

  1. [3]

    Anagnostakis, A

    A. Anagnostakis, A. Lejay, and D. Villemonais. General diffusion processes as limit of time-space Markov chains. Ann. Appl. Probab., 33(5):3620–3651, 2023

    work page 2023

  2. [33]

    Pavlyukevich and A

    I. Pavlyukevich and A. Pilipenko. Walsh’s Brownian motion and Donsker scaling limits of perturbed random walks. ALEA, Lat. Am. J. Probab. Math. Stat. , 21(2):1669–1707, 2024

    work page 2024

  3. [2]

    General diffusions on the star graph as time-changed Walsh Brownian motion

    A. Anagnostakis. General diffusions on the star graph as time-changed Walsh Brownian motion. Preprint, arXiv:2502.19299 [math.PR] (2025), 2025

    work page Pith review arXiv 2025

  4. [1]

    M. Amir. Sticky Brownian motion as the strong limit of a sequence of random walks. Stochastic Process. Appl., 39(2):221–237, 1991

    work page 1991

  5. [4]

    Ankirchner, T

    S. Ankirchner, T. Kruse, W. L¨ ohr, and M. Urusov. Properties of the EMCEL scheme for approximating irregular diffusions. J. Math. Anal. Appl. , 509(1):29, 2022. Id/No 125931

    work page 2022

  6. [5]

    Ankirchner, T

    S. Ankirchner, T. Kruse, and M. Urusov. A functional limit theorem for coin tossing Markov chains. Ann. Inst. Henri Poincar´ e Probab. Stat., 56(4):2996–3019, 2020

    work page 2020

  7. [6]

    Ankirchner, T

    S. Ankirchner, T. Kruse, and M. Urusov. Wasserstein convergence rates for random bit approximations of continuous Markov processes. J. Math. Anal. Appl. , 493(2):31, 2021. Id/No 124543

    work page 2021

  8. [7]

    Barlow, J

    M. Barlow, J. Pitman, and M. Yor. On Walsh’s Brownian motions. S´ eminaire de probabilit´ es XXIII, Lect. Notes Math. 1372, 275-293 (1989)., 1989

    work page 1989

Show all 42 references
  1. [8]

    Berry and F

    J. Berry and F. Camilli. Stationary mean field games on networks with sticky transition conditions. ESAIM, Control Optim. Calc. Var. , 31:23, 2025. Id/No 18

    2025

  2. [9]

    Berry and F

    J. Berry and F. Colantoni. Sticky diffusions on star graphs: characterization and Itˆ o formula. arXiv preprint arXiv:2411.05441 , 2024

    2024

  3. [10]

    Bobrowski and E

    A. Bobrowski and E. Ratajczyk. From snapping out brownian motions to walsh’s spider processes on star-like graphs. arXiv preprint arXiv:2406.16800 , 2024. 37

    2024 arXiv

  4. [11]

    A. N. Borodin and P. Salminen. Handbook of Brownian motion—facts and formulae . Probability and its Applications. Birkh¨ auser Verlag, Basel, 1996

    1996

  5. [12]

    Bou-Rabee and M

    N. Bou-Rabee and M. C. Holmes-Cerfon. Sticky Brownian motion and its numerical solution. SIAM Rev., 62(1):164–195, 2020

    2020

  6. [13]

    Bruggeman and J

    C. Bruggeman and J. Ruf. A one-dimensional diffusion hits points fast. Electron. Commun. Probab., 21:7, 2016. Id/No 22

    2016

  7. [14]

    Cvijovi´ c

    D. Cvijovi´ c. New integral representation of the polylogarithm function.Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. , 463(2080):897–905, 2007

    2007

  8. [15]

    Dassios and J

    A. Dassios and J. Zhang. First hitting time of Brownian motion on simple graph with skew semiaxes. Methodol. Comput. Appl. Probab., 24(3):1805–1831, 2022

    2022

  9. [16]

    M. D. Donsker. An invariance principle for certain probability limit theorems. Mem. Amer. Math. Soc., 6:12, 1951

    1951

  10. [17]

    ´Etor´ e and A

    P. ´Etor´ e and A. Lejay. A Donsker theorem to simulate one-dimensional processes with measurable coefficients. ESAIM Probab. Stat., 11:301–326, 2007

    2007

  11. [18]

    W. Feller. On second order differential operators. Ann. Math. (2) , 61:90–105, 1955

    1955

  12. [19]

    M. I. Freidlin and A. D. Wentzell. Diffusion processes on graphs and the averaging principle. The Annals of probability , pages 2215–2245, 1993

    1993

  13. [20]

    P. K. Friz and N. B. Victoir. Multidimensional stochastic processes as rough paths. Theory and applications., volume 120 of Camb. Stud. Adv. Math. Cambridge: Cambridge University Press, 2010

    2010

  14. [21]

    Itˆ o.Essentials of stochastic processes

    K. Itˆ o.Essentials of stochastic processes. Translated from the 1957 Japanese original. , volume 231 of Transl. Math. Monogr. Providence, RI: American Mathematical Society (AMS), 2006

    1957

  15. [22]

    Itˆ o and H

    K. Itˆ o and H. P. jun. McKean.Diffusion processes and their sample paths. Berlin: Springer- Verlag, 1996

    1996

  16. [23]

    Kallenberg

    O. Kallenberg. Foundations of modern probability. In 2 volumes , volume 99 of Probab. Theory Stoch. Model. Cham: Springer, 3rd revised and expanded edition edition, 2021

    2021

  17. [24]

    A. Lejay. Simulating a diffusion on a graph. Application to reservoir engineering. Monte Carlo Methods Appl., 9(3):241–255, 2003

    2003

  18. [25]

    Lempa, E

    J. Lempa, E. Mordecki, and P. Salminen. Diffusion spiders: Green kernel, excessive functions and optimal stopping. Stochastic Processes and their Applications, 167:104229, 2024

    2024

  19. [26]

    G. Lumer. Connecting of local operators and evolution equations on networks. Potential theory, Proc. Colloq., Copenhagen 1979, Lect. Notes Math. 787, 219-234 (1980)., 1980

    1980

  20. [27]

    P. Mandl. Analytical treatment of one-dimensional Markov processes , volume 151 of Grundlehren Math. Wiss. Berlin-Heidelberg-New York: Springer Verlag, 1968

    1968

  21. [28]

    Martinez and I

    M. Martinez and I. Ohavi. Martingale problem for a Walsh spider process with spinning measure selected from its own local time. Electron. J. Probab., 30:39, 2025. Id/No 22

    2025

  22. [29]

    Meier, L

    C. Meier, L. Li, and G. Zhang. Markov chain approximation of one-dimensional sticky diffusions. Adv. Appl. Probab., 53(2):335–369, 2021

    2021

  23. [30]

    D. Mugnolo. What is actually a metric graph? arXiv preprint arXiv:1912.07549 , 2019

    1912 arXiv

  24. [31]

    I. Ohavi. Comparison principle for Walsh’s spider HJB equations with non linear local-time Kirchhoff’s boundary condition. J. Math. Anal. Appl. , 547(2):35, 2025. Id/No 129300

    2025

  25. [32]

    Ohavi and M

    I. Ohavi and M. Martinez. On spider diffusions having a spinning measure selected from their own local time. Preprint, arXiv:2502.02754 [math.PR] (2025), 2025

    2025 arXiv

  26. [34]

    Revuz and M

    D. Revuz and M. Yor. Continuous martingales and Brownian motion , volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, third edition, 1999. 38

    1999

  27. [35]

    L. C. G. Rogers and D. Williams. Diffusions, Markov processes, and martingales. Vol

  28. [36]

    Cambridge University Press, Cambridge, 2000

    Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2000. Itˆ o calculus, Reprint of the second (1994) edition

    1994

  29. [37]

    T. S. Salisbury. Construction of right processes from excursions. Probab. Theory Relat. Fields, 73:351–367, 1986

    1986

  30. [38]

    Salminen and D

    P. Salminen and D. Stenlund. Occupation times on the legs of a diffusion spider. arXiv preprint arXiv:2411.09976, 2024

    2024 arXiv

  31. [39]

    C. Villani. Optimal transport. Old and new , volume 338 of Grundlehren Math. Wiss. Berlin: Springer, 2009

    2009

  32. [40]

    T. D. Vo and M. Peign´ e. A functional limit theorem for lattice oscillating random walks. ALEA, Lat. Am. J. Probab. Math. Stat. , 20(2):1433–1457, 2023

    2023

  33. [41]

    J. B. Walsh. A diffusion with a discontinuous local time. Ast´ erisque, 52(53):37–45, 1978

    1978

  34. [42]

    M. Weber. On occupation time functionals for diffusion processes and birth-and-death processes on graphs. The Annals of Applied Probability , 11(2):544–567, 2001. 39

    2001

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.