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arxiv: 2604.03030 · v1 · submitted 2026-04-03 · 🧮 math.PR

A localized coupling approach to interacting continuous-state branching processes

Shukai Chen , Pei-Sen Li , Jian Wang This is my paper

Pith reviewed 2026-05-13 17:53 UTC · model grok-4.3

classification 🧮 math.PR MSC 60J8060H1060J25
keywords continuous-state branching processesuniform ergodicitytotal variation convergenceMarkovian couplingstochastic differential equations with jumpsLotka-Volterra interactionsimmigration predation competition
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The pith

Localized Markovian coupling establishes sharp uniform ergodicity conditions for interacting continuous-state branching processes with immigration, predation and competition.

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

The paper constructs a two-dimensional model that merges classical Lotka-Volterra interactions with continuous-state branching processes featuring competition. It represents the system as the unique strong solution of an SDE with jumps and then derives explicit inequalities on the interaction rates that guarantee the process converges in total variation distance to a unique stationary distribution, uniformly in the starting point. The argument rests on a new localized coupling that contracts the distance between two copies of the process only when they are close enough for the jump terms to interact. A reader cares because the result supplies concrete long-run behavior for population models that combine branching, predation, and competition, without requiring global Lipschitz conditions on the coefficients.

Core claim

We establish sharp conditions for the uniform ergodicity in the total variation of this model. Our proof relies on a novel, localized Markovian coupling approach, which is of its own interest in the ergodicity theory of Markov processes with interactions.

What carries the argument

The localized Markovian coupling, which contracts the distance between two coupled copies only inside a neighborhood where the interaction rates satisfy explicit inequalities.

If this is right

  • The process admits a unique stationary distribution and converges to it exponentially fast in total variation.
  • The same localized-coupling technique can be applied to other jump-diffusion models whose interaction structure allows local contraction.
  • Uniform ergodicity holds precisely when the interaction parameters lie inside an explicitly described region of parameter space.
  • The model reduces to known one-dimensional continuous-state branching processes when the second population is removed.

Where Pith is reading between the lines

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

  • The method may extend to systems with more than two interacting populations by iterating the local contraction argument.
  • Numerical simulation of sample paths could test whether the derived inequalities are close to necessary.
  • The coupling construction offers a template for proving ergodicity in other interacting branching or Lévy-driven population models that lack global monotonicity.

Load-bearing premise

The coefficients of the two-dimensional SDE must guarantee a unique strong solution, and the interaction rates must obey the specific inequalities that let the localized coupling contract distances between the coupled processes.

What would settle it

A counter-example process satisfying the stated coefficient conditions yet failing to converge in total variation distance to a single stationary measure from arbitrary initial states.

read the original abstract

We introduce a class of continuous-state branching processes with immigration, predation and competition, which can be viewed as a combination of the classical Lotka-Volterra model and continuous-state branching processes with competition that were introduced by Berestycki, Fittipaldi, and Fontbona (Probab. Theory Relat. Fields, 2018). This model can be constructed as a unique strong solution to a class of two-dimensional stochastic differential equations with jumps. We establish sharp conditions for the uniform ergodicity in the total variation of this model. Our proof relies on a novel, localized Markovian coupling approach, which is of its own interest in the ergodicity theory of Markov processes with interactions.

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.

Referee Report

2 major / 2 minor

Summary. The paper introduces a class of two-dimensional continuous-state branching processes with immigration, predation, and competition, constructed as the unique strong solution to an SDE with jumps. It derives sharp conditions on the coefficients and interaction rates that guarantee uniform ergodicity in total variation distance, with the proof relying on a novel localized Markovian coupling that contracts distances inside a controlled region combined with Foster-Lyapunov arguments outside it.

Significance. If the central claims hold, the work extends ergodicity theory for branching processes to an interacting Lotka-Volterra-type setting and contributes a localized coupling technique of independent interest for Markov processes with interactions. The approach is a direct adaptation rather than a circular construction, and the provision of sharp (rather than sufficient) conditions strengthens the result.

major comments (2)
  1. [§3.2, Theorem 3.4] §3.2, Theorem 3.4: the localized coupling is claimed to yield a contraction rate independent of the starting points once inside the localization ball; however, the explicit dependence of the ball radius on the predation/competition coefficients is not derived, which is load-bearing for the subsequent uniform ergodicity statement.
  2. [§4.1, Eq. (4.3)] §4.1, Eq. (4.3): the Foster-Lyapunov drift condition outside the localized region uses a test function whose growth is linear in the total mass; the verification that this yields a finite return time under the sharp parameter regime requires an additional estimate on the jump intensity that is only sketched.
minor comments (2)
  1. [§2] Notation for the two-dimensional jump measure and the interaction kernel is introduced piecemeal; a consolidated table of symbols in §2 would improve readability.
  2. [Abstract] The abstract states 'sharp conditions' without listing them; adding a one-sentence summary of the precise inequalities on the rates would help readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and constructive comments on our manuscript. We address each major comment below and will incorporate the requested clarifications in the revised version.

read point-by-point responses

  1. Referee: [§3.2, Theorem 3.4] §3.2, Theorem 3.4: the localized coupling is claimed to yield a contraction rate independent of the starting points once inside the localization ball; however, the explicit dependence of the ball radius on the predation/competition coefficients is not derived, which is load-bearing for the subsequent uniform ergodicity statement.

    Authors: We agree that an explicit derivation of the localization ball radius in terms of the predation and competition coefficients will make the argument fully transparent. In the revised manuscript we will add this derivation to the proof of Theorem 3.4 (and the surrounding discussion in §3.2), showing the precise scaling of the radius with the interaction rates that guarantees a contraction rate independent of the starting points inside the ball. This will also clarify its role in obtaining the uniform ergodicity result. revision: yes

  2. Referee: [§4.1, Eq. (4.3)] §4.1, Eq. (4.3): the Foster-Lyapunov drift condition outside the localized region uses a test function whose growth is linear in the total mass; the verification that this yields a finite return time under the sharp parameter regime requires an additional estimate on the jump intensity that is only sketched.

    Authors: We thank the referee for noting that the jump-intensity estimate was only sketched. In the revised Section 4.1 we will expand the argument by supplying the complete, self-contained bound on the jump intensity. This will rigorously verify that the linear-growth test function produces a finite expected return time under the sharp parameter regime, thereby completing the Foster-Lyapunov drift condition without changing the main statements. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper constructs a new class of interacting continuous-state branching processes as the unique strong solution to a two-dimensional jump SDE and proves uniform ergodicity in total variation by introducing a localized Markovian coupling that contracts distance inside a parameter-controlled region, combined with standard Foster-Lyapunov drift arguments outside it. This derivation chain is self-contained: the coupling is built directly from the model coefficients and interaction rates under the stated inequalities, without any reduction to fitted parameters, self-referential definitions, or load-bearing self-citations. Prior work is cited only for the classical Lotka-Volterra and branching-process background, which supplies independent context rather than the central ergodicity result. The proof strategy therefore adds independent content and does not collapse to its inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on the existence of a unique strong solution to the SDE and on parameter inequalities that make the localized coupling contractive; these are domain assumptions standard in the field but not independently verified here.

free parameters (1)
  • interaction coefficients
    Rates for predation, competition, and immigration are general parameters assumed to satisfy explicit inequalities for ergodicity; not numerically fitted in the abstract.
axioms (1)
  • domain assumption Existence and uniqueness of strong solution to the two-dimensional SDE with jumps
    Invoked to construct the process; standard Lipschitz or linear-growth conditions on coefficients are implicitly required.

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Works this paper leans on

52 extracted references · 52 canonical work pages

  1. [1]

    and Sandrić, N

    Arapostathis, A., Pang, G. and Sandrić, N. (2019): Ergodicity of a Lévy-driven SDE arising from multiclass many-server queues. Ann. Appl. Probab. , 29, 1070–1126

    work page 2019

  2. [2]

    and Sandrić, N

    Arapostathis, A., Pang, G. and Sandrić, N. (2022): Subexponential upper and lower bounds in Wasserstein distance for Markov processes. Appl. Math. Optim. , 85, article number 37. 33

    work page 2022

  3. [3]

    (1973): Stochastic Differentialgleichungen: Theorie und Anwendung

    Arnold, L. (1973): Stochastic Differentialgleichungen: Theorie und Anwendung. Olden- bourg, München-Wien

    work page 1973

  4. [4]

    and Yuan, C

    Bao, J., Mao, X., Yin, G. and Yuan, C. (2011): Competitive Lotka–Volterra population dynamics with jumps. Nonlinear Anal., 74, 6601–6616

    work page 2011

  5. [5]

    and Wang, J

    Bao, J. and Wang, J. (2023): Coupling methods and exponential ergodicity for two factor affine processes. Math. Nachr. , 296, 1716–1736

    work page 2023

  6. [6]

    and Pap, G

    Barczy, M., Doring, L., Li, Z. and Pap, G. (2014): Stationarity and ergodicity for an affine two factor model. Adv. Appl. Probab. , 46, 878–898

    work page 2014

  7. [7]

    and Sandholm, W

    Benaïm, M., Hofbauer, J. and Sandholm, W. (2008): Robust permanence and imperma- nence for the stochastic replicator dynamics. J. Biol. Dyn. , 2, 180–195

    work page 2008

  8. [8]

    and Fontbona, J

    Berestycki, J., Fittipaldi, M.C. and Fontbona, J. (2018): Ray-Knight representation of flows of branching processes with competition by pruning of Lévy trees. Probab. Theory Relat. Fields , 172, 725–788

    work page 2018

  9. [9]

    and Hartung, L

    Bovier, A. and Hartung, L. (2023): The speed of invasion in an advancing population. J. Math. Biol. , 87, article number 56

    work page 2023

  10. [10]

    and Méléard, S

    Cattiaux, P. and Méléard, S. (2010): Competitive or weak cooperative stochastic Lotka– Volterra systems conditioned on non-extinction. J. Math. Biol. , 60, 797–829

    work page 2010

  11. [11]

    (2005): Eigenvalues, Inequalities, and Ergodic Theory

    Chen, M.F. (2005): Eigenvalues, Inequalities, and Ergodic Theory. Springer, London

    work page 2005

  12. [12]

    and Li, S

    Chen, M.F. and Li, S. (1989): Coupling methods for multidimensional diffusion processes. Ann. Probab. , 17, 151–177

    work page 1989

  13. [13]

    and Wu, Y

    Chen, X., Tsai, J-C. and Wu, Y. (2017): Long time behavior of solutions of a SIS epi- demiological model. SIAM J. Math. Anal. , 49, 3925–3950

    work page 2017

  14. [14]

    (2013): Measure Theory

    Cohn, D. (2013): Measure Theory. 2nd ed. Springer, New York

    work page 2013

  15. [15]

    and Li, Z

    Dawson, D. and Li, Z. (2006): Skew convolution semigroups and affine Markov processes. Ann. Probab. , 34, 1103–1142

    work page 2006

  16. [16]

    and Li, Z

    Dawson, D. and Li, Z. (2012): Stochastic equations, flows and measure-valued processes. Ann. Probab. , 40, 813–857

    work page 2012

  17. [17]

    and Ma, C

    Duhalde, X., Foucart, C. and Ma, C. (2014): On the hitting times of continuous-state branching processes with immigration. Stochastic Process. Appl. , 124, 4182–4201

    work page 2014

  18. [18]

    (2011): Reflection coupling and Wasserstein contractivity without convexity

    Eberle, A. (2011): Reflection coupling and Wasserstein contractivity without convexity. C. R. Math. Acad. Sci. Paris, 349, 1101–1104

    work page 2011

  19. [19]

    (2016): Reflection couplings and contraction rates for diffusions

    Eberle, A. (2016): Reflection couplings and contraction rates for diffusions. Probab. The- ory Relat. Fields , 166, 851–886

    work page 2016

  20. [20]

    (1951): Diffusion processes in genetics

    Feller, W. (1951): Diffusion processes in genetics. In: Proceedings 2nd Berkeley Symp. Math. Statist. Probab. , 1950, 227–246. Univ. of California Press, Berkeley and Los Angeles. 34

    work page 1951

  21. [21]

    and Li, Z

    Fu, Z. and Li, Z. (2010): Stochastic equations of nonnegative processes with jumps. Stochastic Process. Appl. , 120, 306–330

    work page 2010

  22. [22]

    and Watson, H

    Galton, F. and Watson, H. (1874): On the probability of the extinction of families. J. Anthropol. Inst. Great B. and Ireland , 4, 138–144

    work page

  23. [23]

    and Nguyen, D

    Hening, A. and Nguyen, D. (2018): Coexistence and extinction for stochastic Kolmogorov systems. Ann. Appl. Probab. , 28, 1893–1942

    work page 2018

  24. [24]

    and Chesson, P

    Hening, A., Nguyen, D. and Chesson, P. (2018): A general theory of coexistence and extinction for stochastic ecological communities. J. Math. Biol. , 82, article number 56

    work page 2018

  25. [25]

    and Ungureanu, S

    Hening, A., Nguyen, D., Ta, T. and Ungureanu, S. (2025): Long-term behavior of stochas- tic SIQRS epidemic models. J. Math. Biol. , 90, article number 41

    work page 2025

  26. [26]

    and Scheel, A

    Holzer, M. and Scheel, A. (2014): Accelerated fronts in a two-stage invasion process. SIAM J. Math. Anal. , 46, 397–427

    work page 2014

  27. [27]

    and Watanabe, S

    Ikeda, N. and Watanabe, S. (1989): Stochastic Differential Equations and Diffusion Pro- cesses. North-Holland/Kodansha, Amsterdam/Tokyo

    work page 1989

  28. [28]

    and Rüdiger, B

    Jin, P., Kremer, J. and Rüdiger, B. (2017): Exponential ergodicity of an affine two-factor model based on the α-root process. Adv. Appl. Probab. , 49, 1144–1169

    work page 2017

  29. [29]

    (2005): The branching process with logistic growth

    Lambert, A. (2005): The branching process with logistic growth. Ann. Appl. Probab. , 15, 1506–1535

    work page 2005

  30. [30]

    and Zhou, X

    Li, P., Li, Z., Wang, J. and Zhou, X. (2025): Exponential ergodicity of branching processes with immigration and competition. Ann. Inst. Henri Poincaré Probab. Stat. , 61, 350– 384

    work page 2025

  31. [31]

    and Ma, C

    Li, Z. and Ma, C. (2015): Asymptotic properties of estimators in a stable Cox-Ingersoll- Ross model. Stochastic Process. Appl. , 125, 3196–3233

    work page 2015

  32. [32]

    and Wang, J

    Li, P. and Wang, J. (2020): Exponential ergodicity for general continuous-state nonlinear branching processes. Elect. J. Probab. , 25, 1–25

    work page 2020

  33. [33]

    and Wang, J

    Liang, M., Majka, M. and Wang, J. (2021): Exponential ergodicity for SDEs and McKean- Vlasov processes with Lévy noise. Ann. Inst. Henri Poincaré Probab. Stat. , 57, 1665– 1701

    work page 2021

  34. [34]

    (1992): Lectures on the Coupling Method

    Lindvall, T. (1992): Lectures on the Coupling Method. Wiley, New York

    work page 1992

  35. [35]

    (1920): Analytical note on certain rhythmic relations in organic systems

    Lotka, A. (1920): Analytical note on certain rhythmic relations in organic systems. Proc. Natl. Acad. Sci. , 6, 410–415

    work page 1920

  36. [36]

    and Wang, J

    Luo, D. and Wang, J. (2019): Refined basic couplings and Wasserstein-type distances for SDEs with Lévy noises. Stochastic Process. Appl. , 129, 3129–3173

    work page 2019

  37. [37]

    (2017): Coupling and exponential ergodicity for stochastic differential equa- tions driven by Lévy process

    Majka, M. (2017): Coupling and exponential ergodicity for stochastic differential equa- tions driven by Lévy process. Stochastic Process. Appl. , 127, 4083–4125. 35

    work page 2017

  38. [38]

    (2019): Transportation inequalities for non-globally dissipative SDEs with jumps via Malliavin calculus and coupling

    Majka, M. (2019): Transportation inequalities for non-globally dissipative SDEs with jumps via Malliavin calculus and coupling. Ann. Inst. Henri Poincaré Probab. Stat. , 55, 2019–2057

    work page 2019

  39. [39]

    (2011): Stationary distribution of stochastic population systems

    Mao, X. (2011): Stationary distribution of stochastic population systems. Systems & Cont. Letters , 60, 398–405

    work page 2011

  40. [40]

    (2002): Strong ergodicity for Markov processes by coupling methods

    Mao, Y. (2002): Strong ergodicity for Markov processes by coupling methods. J. Appl. Probab., 39, 839–852

    work page 2002

  41. [41]

    and Villemonais, D

    Méléard, S. and Villemonais, D. (2012): Quasi-stationary distributions and population processes. Probab. Surv. , 9, 340–410

    work page 2012

  42. [42]

    and Tweedie, R

    Meyn, S. and Tweedie, R. (1993): Stability of Markovian processes III: Foster–Lyapunov critera for continuous-time processes. Adv. Appl. Probab. , 25, 518–548

    work page 1993

  43. [43]

    and Yin, G

    Nguyen, D. and Yin, G. (2017): Coexistence and exclusion of stochastic competitive Lotka–Volterra models. J. Differ. Equa. , 262, 1192–1225

    work page 2017

  44. [44]

    and Zhou, X

    Ren, Y., Xiong, J., Yang, X. and Zhou, X. (2022): On the extinction-extinguishing di- chotomy for a stochastic Lotka–Volterra type population dynamical system. Stochastic Process. Appl., 150, 50–90

    work page 2022

  45. [45]

    (2003): Long-time behaviour of a stochastic prey-predator model

    Rudnicki, R. (2003): Long-time behaviour of a stochastic prey-predator model. Stochastic Process. Appl., 108, 93–107

    work page 2003

  46. [46]

    and Wang, J

    Schilling, R. and Wang, J. (2012): On the coupling property and the Liouville theorem for Ornstein-Uhlenbeck processes. J. Evol. Equ., 12, 119–140

    work page 2012

  47. [47]

    (2009): Optimal Transport, Old and New

    Villani, C. (2009): Optimal Transport, Old and New. Springer, Berlin

    work page 2009

  48. [48]

    (1926): Variazioni e fluttuazioni del numero d¡¯individui in specie animali conviventi

    Volterra, V. (1926): Variazioni e fluttuazioni del numero d¡¯individui in specie animali conviventi. Mem. Accad. Lincei, 6, 31–113

    work page 1926

  49. [49]

    (2011): Coupling for Ornstein-Uhlenbeck processes with jumps

    Wang, F.Y. (2011): Coupling for Ornstein-Uhlenbeck processes with jumps. Bernoulli, 17, 1136–1158

    work page 2011

  50. [50]

    and Zhang, X.C

    Xie, L. and Zhang, X.C. (2020): Ergodicity of stochastic differential equations with jumps and singular coefficients. Ann. Inst. Henri Poincaré Probab. Stat. , 56, 175–229

    work page 2020

  51. [51]

    and Zhang, X.L

    Zhang, X.C. and Zhang, X.L. (2023): Ergodicity of supercritical SDEs driven by α-stable processes and heavy-tailed sampling. Bernoulli, 29, 1933-1958

    work page 2023

  52. [52]

    and Yin, G

    Zhu, C. and Yin, G. (2009): On hybird competitive Lotka-Volterra ecosystems. Nonlinear Anal., 71, 1370–1379. 36

    work page 2009