High-dimensional limits for reflected Brownian motion in the orthant

Rami Atar

arxiv: 2605.01958 · v1 · submitted 2026-05-03 · 🧮 math.PR

High-dimensional limits for reflected Brownian motion in the orthant

Rami Atar This is my paper

Pith reviewed 2026-05-09 16:08 UTC · model grok-4.3

classification 🧮 math.PR

keywords propagation of chaosreflected Brownian motionMcKean-Vlasov limitinteracting particlesorthantheavy trafficlocal timeJackson networks

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The pith

A system of n interacting reflected Brownian motions converges to a nonlinear reflected process as n grows large

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

The paper establishes propagation of chaos for an exchangeable collection of Brownian particles reflected at the origin, with each particle's local time at zero entering the dynamics of the others through fixed or random coefficients. Under the stability condition that the average coefficient exceeds negative one, every finite collection of particles converges in law to independent copies of the same nonlinear reflected Brownian motion whose reflection term includes the expected local time of the process. The same McKean-Vlasov limit holds for random heterogeneous coefficients that average to the same value, whether the coefficients are held fixed or averaged over their distribution. This supplies a closed macroscopic equation for the high-dimensional systems that arise as heavy-traffic limits of Jackson networks.

Core claim

When the homogeneous interaction coefficient satisfies a greater than negative one, the n-particle semimartingale reflected Brownian motion in the orthant converges to the nonlinear process defined by the equation bar X(t) equals bar X_0 plus bar W(t) plus bar L(t) plus a times the expectation of bar L(t). The identical limit equation is obtained for random coefficients that are independent across particles, have mean a, and take values in a compact subinterval of (-1,1), both in the quenched and annealed regimes.

What carries the argument

The nonlinear reflected Brownian motion bar X(t) = bar X_0 + bar W(t) + bar L(t) + a E[bar L(t)], which encodes the mean-field interaction through the expected local time

If this is right

The limiting nonlinear equation is identical for constant and for random heterogeneous interaction coefficients
Global existence and uniqueness of the finite-particle system hold under the completely-S condition a greater than negative one
The macroscopic description applies to the heavy-traffic limit of large Jackson networks
Convergence holds simultaneously in the quenched and annealed senses for the random-coefficient model

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The mean-field equation could be solved numerically to obtain approximate stationary distributions for large but finite networks without resolving the full orthant problem
Because particles become independent in the limit, variance reduction or importance sampling methods developed for the nonlinear process may carry over to the original particle system
The averaging-out of random coefficients suggests that similar propagation-of-chaos statements may hold for other classes of reflected diffusions whose reflection directions contain bounded random perturbations

Load-bearing premise

The average interaction strength must exceed negative one so that the reflected system remains well-posed for all time

What would settle it

Simulate the finite-n system for successively larger n with a fixed a greater than negative one and check whether the empirical distribution of any single particle converges in law to the unique solution of the nonlinear reflected equation

read the original abstract

We study interacting Brownian particles on the half-line whose interaction occurs through boundary local times at the origin. The particle system is given by \[ X_i^n(t)=X^n_{0,i}+W_i^n(t)+L_i^n(t) +\frac{1}{n-1}\sum_{j\ne i}\rho^n_{ij}L_j^n(t), \qquad i\in[n],\ t\ge0, \] where the initial conditions are exchangeable, the driving Brownian motions $W_i^n$ are i.i.d., and $L_i^n$ denotes the boundary local time of $X_i^n$ at zero. For each fixed coefficient array $\{\rho^n_{ij}\}$, the system can be viewed as a semimartingale reflected Brownian motion in the orthant. We first consider the homogeneous case $\rho^n_{ij}=a$. In this case, global well-posedness holds under the completely-$\mathcal S$ condition $a>-1$. We prove propagation of chaos under this condition; the subregime $a\in(-1,0]$, in the homogeneous setting, was previously covered as part of the results of \cite{baker2025particle}. The limiting process is the nonlinear reflected Brownian motion \[ \bar X(t)=\bar X_0+\bar W(t)+\bar L(t)+a\mathbb E[\bar L(t)], \qquad t\ge0. \] We also treat heterogeneous random coefficients $\rho^n_{ij}$, assumed to have mean $a$, support in a compact subset of $(-1,1)$, and to be independent across $j$ for each $i$. In both the quenched and annealed settings, the particle system converges to the same McKean--Vlasov limit as in the homogeneous case. The model is motivated by large Jackson networks in heavy traffic.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This extends propagation of chaos for reflected particle systems to the full a > -1 range in the homogeneous case and to random heterogeneous coefficients, but the random case leaves a gap on almost-sure well-posedness of the finite-n systems.

read the letter

The paper fills in the missing subregime a in (-1,0] for the homogeneous reflected particle system and adds the heterogeneous random-coefficient version in both quenched and annealed settings. The limit is the same nonlinear reflected Brownian motion in all cases, with the interaction term a times the expected local time. This is a direct extension of the Baker et al. work and keeps the same completely-S condition for the constant case. The setup is motivated by heavy-traffic Jackson networks, which gives it a concrete applied context without forcing the math into something artificial. The proofs appear to follow standard propagation-of-chaos arguments from the finite-n semimartingale reflected Brownian motion to the McKean-Vlasov limit, and the abstract states the results cleanly for both fixed and random coefficients. That is the solid part: the limit identification itself looks standard once well-posedness is granted. The soft spot is exactly the one the stress-test flags. For the random coefficients, the paper only assumes each ρ^n_ij has mean a and lies in a fixed compact subset of (-1,1), independent across j for each i. It verifies the completely-S property only for the constant matrix. Nothing in the stated assumptions automatically guarantees that almost every realization of the random matrix R^n satisfies the completely-S condition on all principal submatrices. If a positive fraction of realizations fail this, the finite-n system is not well-posed as an SRBM and the quenched convergence claim does not apply to those samples. The paper does not appear to supply a uniform argument that closes this, so the heterogeneous results rest on an unverified step. Readers working on mean-field limits for reflected processes or queueing networks will get the most from it; the homogeneous extension is useful for completing the picture, and the random-coefficient version is the part that could matter for models with varying interactions. It is technically grounded enough and addresses a clear open subcase, so it deserves a serious referee rather than a desk reject. I would send it out, but ask the authors to address the almost-sure well-posedness for the random matrices explicitly in the revision.

Referee Report

1 major / 0 minor

Summary. The paper studies high-dimensional limits of n-particle systems of reflected Brownian motions in the positive orthant, with interactions mediated by boundary local times scaled by coefficients ρ^n_ij. For the homogeneous case ρ^n_ij = a > -1, global well-posedness of the finite-n SRBM is established under the completely-S condition, followed by a proof of propagation of chaos to the nonlinear McKean-Vlasov reflected Brownian motion X̄(t) = X̄_0 + W̄(t) + L̄(t) + a E[L̄(t)]. For heterogeneous random coefficients with mean a, support in a compact subset of (-1,1), and row-wise independence, the paper claims that both quenched and annealed versions of the particle system converge to the same limiting process, with motivation from heavy-traffic limits of large Jackson networks.

Significance. If the central claims hold, the results provide a rigorous extension of propagation-of-chaos theorems to heterogeneous reflected systems, which is relevant for approximating high-dimensional queueing networks. The quenched convergence result, in particular, would allow pathwise limits for random reflection matrices without requiring identical coefficients across particles.

major comments (1)

[Heterogeneous random coefficients] Abstract and heterogeneous-coefficients section: the quenched convergence statement presupposes that, for almost every realization of the random array {ρ^n_ij}, the n-particle system is a well-posed semimartingale reflected Brownian motion. This requires the random reflection matrix R^n (1’s on the diagonal, ρ^n_ij/(n-1) off-diagonal) to be completely-S. The manuscript verifies the completely-S property only for the constant homogeneous matrix with a > -1; no argument is supplied showing that the marginal bounds (mean a, compact support in (-1,1)) and row-wise independence imply that every principal submatrix of R^n admits a strictly positive vector x with R^n x > 0. Because the completely-S condition is not automatically inherited, there exist distributions satisfying the paper’s hypotheses for which a positive fraction of realizations render R^n non-completely-S, making the finite-n SR

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for raising this important point about well-posedness in the heterogeneous case. We address the comment in detail below.

read point-by-point responses

Referee: [Heterogeneous random coefficients] Abstract and heterogeneous-coefficients section: the quenched convergence statement presupposes that, for almost every realization of the random array {ρ^n_ij}, the n-particle system is a well-posed semimartingale reflected Brownian motion. This requires the random reflection matrix R^n (1’s on the diagonal, ρ^n_ij/(n-1) off-diagonal) to be completely-S. The manuscript verifies the completely-S property only for the constant homogeneous matrix with a > -1; no argument is supplied showing that the marginal bounds (mean a, compact support in (-1,1)) and row-wise independence imply that every principal submatrix of R^n admits a strictly positive vector x with R^n x > 0. Because the completely-S condition is not automatically inherited, there exist distributions satisfying the paper’s hypotheses for which a positive fraction of realizations render R^n non

Authors: We appreciate the referee drawing attention to this detail. In fact, the completely-S condition holds deterministically (hence almost surely) for every realization under the paper's hypotheses on the random coefficients. Let K denote the compact support of each ρ^n_ij; by assumption K is a compact subset of (-1,1), so there exists M < 1 with |ρ| ≤ M for all ρ ∈ K. Consider an arbitrary principal submatrix R' of R^n of size k ≤ n. Let x be the vector of all ones. For each row i of R', (R'x)_i = 1 + ∑_{j≠i} ρ_ij/(n-1) ≥ 1 − (k−1)M/(n−1) ≥ 1 − (n−1)M/(n−1) = 1 − M > 0. Thus R'x > 0, so every principal submatrix satisfies the defining property of completely-S. The row-wise independence and the value of the mean a are not required for this conclusion; the uniform bound away from ±1 supplied by compactness is sufficient. We will insert a short lemma or remark containing this argument in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: propagation of chaos derived from finite-n SRBM under stated assumptions

full rationale

The paper defines the n-particle system explicitly via the given semimartingale equation with local times, invokes the completely-S condition to assert global well-posedness for the homogeneous case (a > -1), and then applies standard propagation-of-chaos arguments to obtain convergence to the McKean-Vlasov nonlinear reflected Brownian motion whose equation is written with the expectation term. For the heterogeneous case the same limit is recovered under the given marginal conditions on ρ^n_ij without any parameter fitting, self-referential definition, or load-bearing self-citation that reduces the claim to its own inputs. The derivation chain therefore remains independent of the target result and does not collapse by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Relies on standard stochastic analysis assumptions for existence of reflected SDE solutions and propagation of chaos arguments.

axioms (1)

domain assumption Existence and uniqueness of solutions to the reflected particle system under a > -1.
Stated to guarantee the finite-n system is well-defined.

pith-pipeline@v0.9.0 · 9603 in / 821 out tokens · 87340 ms · 2026-05-09T16:08:56.434183+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

[1]

Atar and G

R. Atar and G. Wolansky. Invariance principle and McKean–Vlasov limit for randomized load balancing in heavy traffic.The Annals of Applied Probability, 35(5):3046–3085, 2025

work page 2025
[2]

Baccelli, F

F. Baccelli, F. I. Karpelevich, M. Y. Kelbert, A. A. Puhalskii, A. Rybko, and Y. M. Suhov. A mean-field limit for a class of queueing networks.Journal of statistical physics, 66(3):803–825, 1992

work page 1992
[3]

Baker, B

G. Baker, B. Hambly, and P. Jettkant. Particle systems and McKean–Vlasov dynamics with singular inter- action through local times.arXiv preprint arXiv:2503.08837, 2025

work page arXiv 2025
[4]

C. L. Barnes. Hydrodynamic limit and propagation of chaos for Brownian particles reflecting from a New- tonian barrier.The Annals of Applied Probability, 30(4):1582–1613, 2020. 19

work page 2020
[5]

Bayraktar, A

E. Bayraktar, A. Budhiraja, and A. Cohen. Rate control under heavy traffic with strategic servers.Ann. Appl. Probab., 29(1):1–35, 2019

work page 2019
[6]

K. L. Chung and R. J. Williams.Introduction to Stochastic Integration, volume 2. Springer, 1990

work page 1990
[7]

Coghi, W

M. Coghi, W. Dreyer, P. K. Friz, P. Gajewski, C. Guhlke, and M. Maurelli. A McKean–Vlasov SDE and particle system with interaction from reflecting boundaries.SIAM Journal on Mathematical Analysis, 54 (2):2251–2294, 2022

work page 2022
[8]

G¨ artner

J. G¨ artner. On the McKean-Vlasov limit for interacting diffusions.Mathematische Nachrichten, 137(1): 197–248, 1988

work page 1988
[9]

J. M. Harrison and M. I. Reiman. Reflected Brownian motion on an orthant.Ann. Probab., 9(2):302–308,

work page
[10]

P. D. Hinds, A. Sharma, and M. V. Tretyakov. Well-posedness and approximation of reflected McKean– Vlasov SDEs with applications.Mathematical Models and Methods in Applied Sciences, 35(08):1845–1887, 2025

work page 2025
[11]

H. P. McKean Jr. A class of Markov processes associated with nonlinear parabolic equations.Proceedings of the National Academy of Sciences, 56(6):1907–1911, 1966

work page 1907
[12]

J. L. Menaldi. Stochastic variational inequality for reflected diffusion.Indiana University mathematics journal, 32(5):733–744, 1983

work page 1983
[13]

M. I. Reiman. Open queueing networks in heavy traffic.Mathematics of Operations Research, 9(3):441–458, 1984

work page 1984
[14]

M. I. Reiman and R. J. Williams. A boundary property of semimartingale reflecting Brownian motions. Probability Theory and Related Fields, 77(1):87–97, 1988

work page 1988
[15]

Rybko and S

A. Rybko and S. Shlosman. Stationary states of the generalized Jackson networks.Markov Processes and Related Fields, 22(4):759–774, 2016

work page 2016
[16]

A. N. Rybko and S. B. Shlosman. Poisson hypothesis for information networks. I.Moscow mathematical journal, 5(3):679–704, 2005

work page 2005
[17]

A. N. Rybko and S. B. Shlosman. Poisson hypothesis for information networks. II.Moscow mathematical journal, 5(4):927–959, 2005

work page 2005
[18]

Sznitman

A.-S. Sznitman. Nonlinear reflecting diffusion process, and the propagation of chaos and fluctuations asso- ciated.Journal of Functional Analysis, 56(3):311–336, 1984

work page 1984
[19]

Sznitman

A.-S. Sznitman. Topics in propagation of chaos.Lecture Notes in Mathematics, (1464):165–251, 1991

work page 1991
[20]

L. M. Taylor and R. J. Williams. Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant.Probability Theory and Related Fields, 96(3):283–317, 1993

work page 1993
[21]

R. Wei, S. Yang, and J. Zhai. McKean–Vlasov stochastic differential equations with oblique reflection on non-smooth time-dependent domains.Stochastics, pages 1–27, 2026

work page 2026
[22]

R. J. Williams. An invariance principle for semimartingale reflecting Brownian motions in an orthant. Queueing Systems, 30:5–25, 1998. Viterbi F aculty of Electrical & Computer Engineering, Technion–Israel Institute of Tech- nology Email address:rami@technion.ac.il

work page 1998

[1] [1]

Atar and G

R. Atar and G. Wolansky. Invariance principle and McKean–Vlasov limit for randomized load balancing in heavy traffic.The Annals of Applied Probability, 35(5):3046–3085, 2025

work page 2025

[2] [2]

Baccelli, F

F. Baccelli, F. I. Karpelevich, M. Y. Kelbert, A. A. Puhalskii, A. Rybko, and Y. M. Suhov. A mean-field limit for a class of queueing networks.Journal of statistical physics, 66(3):803–825, 1992

work page 1992

[3] [3]

Baker, B

G. Baker, B. Hambly, and P. Jettkant. Particle systems and McKean–Vlasov dynamics with singular inter- action through local times.arXiv preprint arXiv:2503.08837, 2025

work page arXiv 2025

[4] [4]

C. L. Barnes. Hydrodynamic limit and propagation of chaos for Brownian particles reflecting from a New- tonian barrier.The Annals of Applied Probability, 30(4):1582–1613, 2020. 19

work page 2020

[5] [5]

Bayraktar, A

E. Bayraktar, A. Budhiraja, and A. Cohen. Rate control under heavy traffic with strategic servers.Ann. Appl. Probab., 29(1):1–35, 2019

work page 2019

[6] [6]

K. L. Chung and R. J. Williams.Introduction to Stochastic Integration, volume 2. Springer, 1990

work page 1990

[7] [7]

Coghi, W

M. Coghi, W. Dreyer, P. K. Friz, P. Gajewski, C. Guhlke, and M. Maurelli. A McKean–Vlasov SDE and particle system with interaction from reflecting boundaries.SIAM Journal on Mathematical Analysis, 54 (2):2251–2294, 2022

work page 2022

[8] [8]

G¨ artner

J. G¨ artner. On the McKean-Vlasov limit for interacting diffusions.Mathematische Nachrichten, 137(1): 197–248, 1988

work page 1988

[9] [9]

J. M. Harrison and M. I. Reiman. Reflected Brownian motion on an orthant.Ann. Probab., 9(2):302–308,

work page

[10] [10]

P. D. Hinds, A. Sharma, and M. V. Tretyakov. Well-posedness and approximation of reflected McKean– Vlasov SDEs with applications.Mathematical Models and Methods in Applied Sciences, 35(08):1845–1887, 2025

work page 2025

[11] [11]

H. P. McKean Jr. A class of Markov processes associated with nonlinear parabolic equations.Proceedings of the National Academy of Sciences, 56(6):1907–1911, 1966

work page 1907

[12] [12]

J. L. Menaldi. Stochastic variational inequality for reflected diffusion.Indiana University mathematics journal, 32(5):733–744, 1983

work page 1983

[13] [13]

M. I. Reiman. Open queueing networks in heavy traffic.Mathematics of Operations Research, 9(3):441–458, 1984

work page 1984

[14] [14]

M. I. Reiman and R. J. Williams. A boundary property of semimartingale reflecting Brownian motions. Probability Theory and Related Fields, 77(1):87–97, 1988

work page 1988

[15] [15]

Rybko and S

A. Rybko and S. Shlosman. Stationary states of the generalized Jackson networks.Markov Processes and Related Fields, 22(4):759–774, 2016

work page 2016

[16] [16]

A. N. Rybko and S. B. Shlosman. Poisson hypothesis for information networks. I.Moscow mathematical journal, 5(3):679–704, 2005

work page 2005

[17] [17]

A. N. Rybko and S. B. Shlosman. Poisson hypothesis for information networks. II.Moscow mathematical journal, 5(4):927–959, 2005

work page 2005

[18] [18]

Sznitman

A.-S. Sznitman. Nonlinear reflecting diffusion process, and the propagation of chaos and fluctuations asso- ciated.Journal of Functional Analysis, 56(3):311–336, 1984

work page 1984

[19] [19]

Sznitman

A.-S. Sznitman. Topics in propagation of chaos.Lecture Notes in Mathematics, (1464):165–251, 1991

work page 1991

[20] [20]

L. M. Taylor and R. J. Williams. Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant.Probability Theory and Related Fields, 96(3):283–317, 1993

work page 1993

[21] [21]

R. Wei, S. Yang, and J. Zhai. McKean–Vlasov stochastic differential equations with oblique reflection on non-smooth time-dependent domains.Stochastics, pages 1–27, 2026

work page 2026

[22] [22]

R. J. Williams. An invariance principle for semimartingale reflecting Brownian motions in an orthant. Queueing Systems, 30:5–25, 1998. Viterbi F aculty of Electrical & Computer Engineering, Technion–Israel Institute of Tech- nology Email address:rami@technion.ac.il

work page 1998