From stochastic individual-based models to free-boundary Hamilton-Jacobi equations

Nicolas Champagnat; Sepideh Mirrahimi; Sylvie M\'el\'eard; Viet Chi Tran

arxiv: 2604.06516 · v2 · submitted 2026-04-07 · 🧮 math.PR · math.AP

From stochastic individual-based models to free-boundary Hamilton-Jacobi equations

Nicolas Champagnat , Sylvie M\'el\'eard , Sepideh Mirrahimi , Viet Chi Tran This is my paper

Pith reviewed 2026-05-10 18:08 UTC · model grok-4.3

classification 🧮 math.PR math.AP

keywords stochastic branching modelsfree-boundary Hamilton-Jacobi equationsstate constraintslarge deviationsbranching processesphenotypic traitsextinctionpopulation dynamics

0 comments

The pith

Stochastic branching models for trait-structured populations converge to free-boundary Hamilton-Jacobi equations with state constraints that enforce local extinctions.

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

The paper establishes that a stochastic individual-based branching model for populations structured by a quantitative phenotypic trait, with births, deaths, and mutations, converges in a suitable scaling limit to free-boundary Hamilton-Jacobi equations with state constraints. This limit is taken for large population size, small mutation rates, and on logarithmic scales of size and time. Unlike classical Hamilton-Jacobi equations from deterministic models, the free-boundary versions prevent positive population density in regions of trait space where extinction occurs. A sympathetic reader would care because this supplies a macroscopic description of evolutionary dynamics that incorporates stochastic extinction effects, which can change which traits are predicted to persist.

Core claim

In the regime of large populations and small mutations analyzed in logarithmic scales, the stochastic individual-based system converges to a class of free-boundary Hamilton-Jacobi equations with state constraints. These equations go beyond classical Hamilton-Jacobi equations from deterministic models by accounting for possible extinction of the system in certain regions of the trait space. The derivation combines analysis of Hamilton-Jacobi equations with large deviation principles and tools from branching process theory.

What carries the argument

Free-boundary Hamilton-Jacobi equations with state constraints, obtained by large-deviation analysis of the branching process, which enforce zero population density outside the surviving trait regions.

If this is right

The limiting population density must be zero in any trait region where extinction is favored, creating a sharp free boundary.
The support of the limiting population distribution is determined by the solution of the constrained equations rather than filling the entire trait space.
Evolutionary outcomes can be predicted with explicit inclusion of stochastic die-off in suboptimal trait regions.
The macroscopic model distinguishes persistence from extinction more accurately than interior Hamilton-Jacobi equations alone.

Where Pith is reading between the lines

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

The same scaling approach may apply to models with different mutation kernels or interaction structures, potentially yielding similar free boundaries.
Numerical checks of the limiting support against finite-population simulations could quantify how well the state constraint approximates extinction for moderately large populations.
This limit suggests that deterministic models without state constraints may overestimate long-term survival in marginal trait regions.

Load-bearing premise

The analysis requires a regime of large population size and small mutations, taken specifically in logarithmic scales of size and time.

What would settle it

A numerical simulation of the individual-based branching model in a parameter regime with extinction in part of the trait space, followed by comparison of the simulated population support against the zero level set of the limiting equation, would test whether the free boundary appears as predicted.

read the original abstract

We study a stochastic branching model for a population structured by a quantitative phenotypic trait and subject to births, deaths, and mutations. In a regime of large population and small mutations, and in logarithmic scales of size and time, we derive a certain class of free boundary Hamilton-Jacobi equations with state constraints from the stochastic individual-based system. This goes beyond the classical Hamilton-Jacobi equations obtained from deterministic models by taking into account the possible extinction of the system in certain regions of the trait space. The proof is obtained by combining methods for the analysis of Hamilton-Jacobi equations with probabilistic tools from the theory of large deviations and branching processes.

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 derives free-boundary HJ equations with state constraints from a stochastic branching model, capturing extinction beyond standard deterministic limits.

read the letter

The main point is that the authors start from a stochastic individual-based branching model for trait-structured populations and derive a class of free-boundary Hamilton-Jacobi equations with state constraints in the large-population, small-mutation, logarithmic regime. The state constraints let the limit reflect possible extinction in parts of trait space, which classical deterministic derivations do not capture directly. They reach this by combining large deviation principles, branching process tools, and standard HJ analysis methods. That combination is a reasonable way to keep the probabilistic structure visible in the limit equation. The approach looks consistent and avoids circularity or invented parameters. It builds on established techniques without obvious gaps in the stated strategy. A minor limitation is the specific scaling regime, which is standard for these derivations but restricts direct use to large populations. The abstract outlines the steps clearly enough to see the logic, though a referee would still need to verify the detailed estimates and how the constraints are enforced in the limit. This work is for people working on stochastic models in mathematical biology or evolutionary dynamics who care about asymptotic limits that include extinction. It shows clear engagement with the relevant literature and tools. I would send it for peer review.

Referee Report

0 major / 2 minor

Summary. The manuscript derives a class of free-boundary Hamilton-Jacobi equations with state constraints from a stochastic individual-based branching model for a population structured by a quantitative phenotypic trait. In the regime of large population and small mutations, on logarithmic scales of size and time, the authors combine methods from Hamilton-Jacobi equations with large deviations and branching processes to obtain the limit, accounting for extinction in certain trait regions beyond classical deterministic models.

Significance. If the derivation holds, this result is significant as it provides a rigorous passage from stochastic microscopic models to macroscopic free-boundary PDEs that incorporate the possibility of local extinction, which is a key feature not captured by deterministic approximations. The probabilistic approach using large deviations and branching processes adds robustness to the analysis in the logarithmic regime.

minor comments (2)

[Abstract] The abstract describes the proof strategy at a high level; adding one sentence on the main technical steps (e.g., how the state constraint arising from extinction is obtained via branching-process estimates) would improve accessibility.
Notation for the trait space, the free boundary, and the logarithmic scaling should be introduced with a short table or explicit list in the introduction to aid readers following the large-deviation arguments.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the accurate summary of its contributions, and the recommendation for minor revision. The referee correctly identifies the key novelty in incorporating extinction effects via the stochastic branching model and the use of large-deviation and branching-process techniques in the logarithmic regime.

Circularity Check

0 steps flagged

Derivation from stochastic model to PDE limit is self-contained

full rationale

The paper starts from an explicit stochastic individual-based branching process with births, deaths, and mutations. It invokes standard large-deviation and branching-process tools to pass to the logarithmic large-population/small-mutation limit, obtaining a free-boundary Hamilton-Jacobi equation with state constraints. No equation or step is shown to be presupposed by the target PDE, no parameter is fitted to data and then relabeled a prediction, and no load-bearing uniqueness result is imported solely via self-citation. The derivation therefore remains independent of its own output.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The derivation rests on standard asymptotic assumptions in the field of stochastic processes and PDE limits, which are domain assumptions rather than ad hoc inventions.

axioms (4)

domain assumption The population is large
Required for the mean-field like limit
domain assumption Mutations are small
Enables the specific scaling
domain assumption Logarithmic scaling in size and time
Used to capture the asymptotic behavior
domain assumption The model is a branching process with births, deaths, mutations
The starting stochastic individual-based model

pith-pipeline@v0.9.0 · 5409 in / 1469 out tokens · 65093 ms · 2026-05-10T18:08:52.961930+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel (J uniqueness); branch_selection (coupling excludes additive branch) echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

ua(t,x)=sup{Ft(f):f∈AC[0,t],f(t)=x,∀s Fs(f)≥a} ... state-constrained Hamilton-Jacobi ... Ωa open ... ∂tu=p(x)H(∂xu)+R(x) on Ωa, u=a on ∂Ωa
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability; bare_distinguishability_of_absolute_floor echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

only paths fs such that Fs(f)>0 for all s∈[0,t] are admissible: the population gets extinct on the way otherwise

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

46 extracted references · 46 canonical work pages

[1]

Bansaye, J.F

V. Bansaye, J.F. Delmas, L. Marsalle and V.C. Tran. Limit theorems for Markov processes indexed by continuous time Galton-Watson trees. The Annals of Applied Probability, Vol. 21, No. 6, 2263-2314, (2011)

work page 2011
[2]

G. Barles. Solutions de viscosit´ edes ´ equationsde Hamilton-Jacobi. Springer-Verlag Berlin Heidelberg, 1994. [3]G. Barles,An Introduction to the Theory of Viscosity Solutions for First-Order Hamilton-Jacobi Equations and Applications,Lecture Notes in Mathematics 2074, 2013

work page 1994
[3]

Barles, L

G. Barles, L. C. Evans, and P. E. Souganidis. Wavefront propagation for reaction-diffusion systems of PDE. Duke Math. J., 61(3):835–858, 1990

work page 1990
[4]

Barles, S

G. Barles, S. Mirrahimi, and B. Perthame. Concentration in Lotka-Volterra parabolic or integral equations: a general convergence result. Methods and Applications of Analysis, 16(3):321–340, 2009

work page 2009
[5]

Berestycki and E

J. Berestycki and E. Brunet and J.W. Harris and J. W. Harris and C. Simon and M.I. Roberts. Growth rates of the population in a branching Brownian motion with an inhomogeneous breeding potential. Stochastic Process. Appl. 125 (2015), no. 5, 2096–2145

work page 2015
[6]

J.D. Biggins. The growth and spread of the general branching random walk. The Annals of Applied Probability, 5(4):1008-1024, (1995)

work page 1995
[7]

Billingsley

P. Billingsley. Convergence of Probability Measures. John Wiley & Sons, New York (1968)

work page 1968
[8]

Bovier, L

A. Bovier, L. Coquille, C. Smadi. Crossing a fitness valley as a metastable transition in a stochastic population model. Ann. Appl. Probab. 29(6), 3541–3589 (2019). 42

work page 2019
[9]

Calvez, B

V. Calvez, B. Henry, S. M´ el´ eard, V.C. Tran. Dynamics of lineages in adaptation to a gradual environmental change. Annales Henri Lebesgue, 5:729–777, 2022. [11]V. Calvez, V. and K.-Y. Lam,Uniqueness of the viscosity solution of a constrained Hamilton-Jacobi equation, Calc. Var. Partial Differ. Equ.,59(5)(2020) pp. 163

work page 2022
[10]

Champagnat

N. Champagnat. A microscopic interpretation for adaptative dynamics trait substitution sequence models. Stochastic Processes and their Applications, 116:1127–1160, 2006

work page 2006
[11]

Champagnat, R

N. Champagnat, R. Ferri` ere, and S. M´ el´ eard. Unifying evolutionary dynamics: from individual stochastic processes to macroscopic models via timescale separation. Theoretical Population Biology, 69:297–321, 2006

work page 2006
[12]

Champagnat and S

N. Champagnat and S. M´ el´ eard. Polymorphic evolution sequence and evolutionary branching. Probability Theory and Related Fields, 151(1-2):45–94, 2011

work page 2011
[13]

Champagnat, S

N. Champagnat, S. M´ el´ eard and V.C. Tran. Stochastic analysis of emergence of evolutionary cyclic behavior in population dynamics with transfer. Ann. Appl. Probab., 31(4), 1820–1867, 2021

work page 2021
[14]

Champagnat, S

N. Champagnat, S. M´ el´ eard, S. Mirrahimi and V.C. Tran. Filling the gap between individual-based evolu- tionary models and Hamilton-Jacobi equations. J. Ec. Polytechnique, 10, 1247–1275, 2023

work page 2023
[15]

Coquille and A

L. Coquille and A. Kraut and C. Smadi. Stochastic individual-based models with power law mutation rate on a general finite trait space. Electron. J. Probab., 26, 1–37, 2021. [18]G. Dal Maso, H. Frankowska,Value functions for Bolza problems with discontinuous Lagrangians and Hamilton-Jacobi inequalities, ESAIM: Control, Optimisation and Calculus of Variati...

work page 2021
[16]

Dembo and O

A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications. Vol 38., Springer, second edition, 1998

work page 1998
[17]

Diekmann, P.-E

O. Diekmann, P.-E. Jabin, S. Mischler, and B. Perthame. The dynamics of adaptation: an illuminating example and a Hamilton-Jacobi approach. Theoretical Population Biology, 67, 257–271, 2005

work page 2005
[18]

Dupuis and R.S

P. Dupuis and R.S. Ellis. A Weak Convergence Approach to the Theory of Large Deviations. Wiley Series in Probability and Statistics, 1997

work page 1997
[19]

Durrett and J

R. Durrett and J. Mayberry. Travelling waves of selective sweeps. Annals of Applied Probability, 21(2), 699–744, 2011

work page 2011
[20]

Esser and A

M. Esser and A. Kraut. A general multi-scale description of metastable adaptive motion across fitness valleys. J. Math. Biol., 89(46), 2024

work page 2024
[21]

Ethier and Thomas G, Kurtz

Stewart N. Ethier and Thomas G, Kurtz. Markov processes: characterization and convergence. John Wiley & Sons, 1986

work page 1986
[22]

L. C. Evans Partial differential equations Graduate Studies in Mathematics Vol. 19, American Mathematical Society, 1998

work page 1998
[23]

L. C. Evans and P. E. Souganidis. A PDE approach to geometric optics for certain semilinear parabolic equations. Indiana Univ. Math. J., 38(1):141–172, 1989

work page 1989
[24]

Fang and O

M. Fang and O. Zeitouni. Slowdown for Time Inhomogeneous Branching Brownian Motion. J. Statistical Physics, 149:1-9, (2012)

work page 2012
[25]

A. Fathi. Weak Kam Theorem in Lagrangian Dynamics. Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2016

work page 2016
[26]

Limit theorems for large deviations and reaction-diffusion equations

M Freidlin. Limit theorems for large deviations and reaction-diffusion equations. The Annals of Probability, 13(3):639–675, 1985

work page 1985
[27]

Forien, J

R. Forien, J. Garnier and F. Patout. Ancestral lineages in mutation-selection equilibria with moving optimum. Bull Math Biol., 84(93), 2022

work page 2022
[28]

Fournier and S

N. Fournier and S. M´ el´ eard. A Microscopic Probabilistic Description of a Locally Regulated Population and Macroscopic Approximations. Ann. Appl. Probab., 14(4):1880-1919, 2004

work page 1919
[29]

Hardy and S.C

R. Hardy and S.C. Harris. A Spine Approach to Branching Diffusions with Applications to Lp-Convergence of Martingales. S´ eminairede Probabilit´ esXLII, Lecture Notes in Mathematics, 1979:281-330, Springer Nature, 2009

work page 1979
[30]

Henry, S

B. Henry, S. M´ el´ eard and V.C. Tran. Time reversal of spinal processes for linear and non-linear branching processes near stationarity. Electronic Journal of Probability, 28(32):1–27, 2023. 43

work page 2023
[31]

Ikeda and S

N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes, North-Holland Pub- lishing Company, 1989

work page 1989
[32]

P.-E. Jabin. Small populations corrections for selection-mutation models. Netw. Heterog. Media, 7(4), 805–836, 2012

work page 2012
[33]

A. Jeddi. Asymptotic behavior of some stochastic models in population dynamics: a Hamilton-Jacobi approach. ArXiV arXiv:2602.20825, 2026

work page arXiv 2026
[34]

Last and M

G. Last and M. Penrose. Lectures on the Poisson Process. IMS Textbook by Cambridge University Press (2017)

work page 2017
[35]

A. Lorz, S. Mirrahimi and B Perthame. Dirac mass dynamics in a multidimensional nonlocal parabolic equation. Communications in Partial Differential Equations, 36, 1071–1098 (2011)

work page 2011
[36]

Maillard and G

P. Maillard and G. Raoul and J. Tourniaire. Spreading speed of locally regulated population models in macroscopically heterogeneous environments (2024). ArXiv:2105.06985

work page arXiv 2024
[37]

B. Mallein. Maximal displacement of a branching random walk in time- inhomogeneous environment. Stochastic Processes and Their Applications, 125(10), 3958–4019 (2015)

work page 2015
[38]

A. Marguet. Uniform sampling in a structured branching population. Bernoulli, 25, 4A, 2649-2695, 2019

work page 2019
[39]

M´ el´ eard and V.C

S. M´ el´ eard and V.C. Tran. Nonlinear historical superprocess approximations for population models with past dependence. Electronic Journal of Probability, 17(47):1-32, 2012

work page 2012
[40]

Mirrahimi, G

S. Mirrahimi, G. Barles, B. Perthame, P.E. Souganidis. A singular Hamilton-Jacobi equation modeling the tail problem. SIAM J. Math. Anal., 44 (6), 4297–4319, 2012

work page 2012
[41]

Perthame and G

B. Perthame and G. Barles. Dirac concentrations in Lotka-Volterra parabolic PDEs. Indiana Univ. Math. J. 57, 3275–3301 (2008)

work page 2008
[42]

Perthame and M

B. Perthame and M. Gauduchon. Survival thresholds and mortality rates in adaptive dynamics: conciliating deterministic and stochastic simulations. Math. Med. Biol. 27(3), 195–210 (2010)

work page 2010
[43]

Protter.Stochastic integration and differential equations, second Edition

P.E. Protter.Stochastic integration and differential equations, second Edition. Stochastic Modelling and Applied Probability, 21, Springer, Berlin, 2004

work page 2004
[44]

Desmarais, E

C. Desmarais, E. Schertzer, Z. Talygi´ as. K-Branching random walk with noisy selection: large population limits and phase transitions. arXiv:2509.26254

work page arXiv
[45]

Waxman and S

D. Waxman and S. Gavrilets. 20 Questions on Adaptive Dynamics. Journal of Evolutionary Biology 18, 1139-1154 (2005)

work page 2005
[46]

Zeitouni

O. Zeitouni. Branching random walks and Gaussian fields. Lecture notes, (2012). 44

work page 2012

[1] [1]

Bansaye, J.F

V. Bansaye, J.F. Delmas, L. Marsalle and V.C. Tran. Limit theorems for Markov processes indexed by continuous time Galton-Watson trees. The Annals of Applied Probability, Vol. 21, No. 6, 2263-2314, (2011)

work page 2011

[2] [2]

G. Barles. Solutions de viscosit´ edes ´ equationsde Hamilton-Jacobi. Springer-Verlag Berlin Heidelberg, 1994. [3]G. Barles,An Introduction to the Theory of Viscosity Solutions for First-Order Hamilton-Jacobi Equations and Applications,Lecture Notes in Mathematics 2074, 2013

work page 1994

[3] [3]

Barles, L

G. Barles, L. C. Evans, and P. E. Souganidis. Wavefront propagation for reaction-diffusion systems of PDE. Duke Math. J., 61(3):835–858, 1990

work page 1990

[4] [4]

Barles, S

G. Barles, S. Mirrahimi, and B. Perthame. Concentration in Lotka-Volterra parabolic or integral equations: a general convergence result. Methods and Applications of Analysis, 16(3):321–340, 2009

work page 2009

[5] [5]

Berestycki and E

J. Berestycki and E. Brunet and J.W. Harris and J. W. Harris and C. Simon and M.I. Roberts. Growth rates of the population in a branching Brownian motion with an inhomogeneous breeding potential. Stochastic Process. Appl. 125 (2015), no. 5, 2096–2145

work page 2015

[6] [6]

J.D. Biggins. The growth and spread of the general branching random walk. The Annals of Applied Probability, 5(4):1008-1024, (1995)

work page 1995

[7] [7]

Billingsley

P. Billingsley. Convergence of Probability Measures. John Wiley & Sons, New York (1968)

work page 1968

[8] [8]

Bovier, L

A. Bovier, L. Coquille, C. Smadi. Crossing a fitness valley as a metastable transition in a stochastic population model. Ann. Appl. Probab. 29(6), 3541–3589 (2019). 42

work page 2019

[9] [9]

Calvez, B

V. Calvez, B. Henry, S. M´ el´ eard, V.C. Tran. Dynamics of lineages in adaptation to a gradual environmental change. Annales Henri Lebesgue, 5:729–777, 2022. [11]V. Calvez, V. and K.-Y. Lam,Uniqueness of the viscosity solution of a constrained Hamilton-Jacobi equation, Calc. Var. Partial Differ. Equ.,59(5)(2020) pp. 163

work page 2022

[10] [10]

Champagnat

N. Champagnat. A microscopic interpretation for adaptative dynamics trait substitution sequence models. Stochastic Processes and their Applications, 116:1127–1160, 2006

work page 2006

[11] [11]

Champagnat, R

N. Champagnat, R. Ferri` ere, and S. M´ el´ eard. Unifying evolutionary dynamics: from individual stochastic processes to macroscopic models via timescale separation. Theoretical Population Biology, 69:297–321, 2006

work page 2006

[12] [12]

Champagnat and S

N. Champagnat and S. M´ el´ eard. Polymorphic evolution sequence and evolutionary branching. Probability Theory and Related Fields, 151(1-2):45–94, 2011

work page 2011

[13] [13]

Champagnat, S

N. Champagnat, S. M´ el´ eard and V.C. Tran. Stochastic analysis of emergence of evolutionary cyclic behavior in population dynamics with transfer. Ann. Appl. Probab., 31(4), 1820–1867, 2021

work page 2021

[14] [14]

Champagnat, S

N. Champagnat, S. M´ el´ eard, S. Mirrahimi and V.C. Tran. Filling the gap between individual-based evolu- tionary models and Hamilton-Jacobi equations. J. Ec. Polytechnique, 10, 1247–1275, 2023

work page 2023

[15] [15]

Coquille and A

L. Coquille and A. Kraut and C. Smadi. Stochastic individual-based models with power law mutation rate on a general finite trait space. Electron. J. Probab., 26, 1–37, 2021. [18]G. Dal Maso, H. Frankowska,Value functions for Bolza problems with discontinuous Lagrangians and Hamilton-Jacobi inequalities, ESAIM: Control, Optimisation and Calculus of Variati...

work page 2021

[16] [16]

Dembo and O

A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications. Vol 38., Springer, second edition, 1998

work page 1998

[17] [17]

Diekmann, P.-E

O. Diekmann, P.-E. Jabin, S. Mischler, and B. Perthame. The dynamics of adaptation: an illuminating example and a Hamilton-Jacobi approach. Theoretical Population Biology, 67, 257–271, 2005

work page 2005

[18] [18]

Dupuis and R.S

P. Dupuis and R.S. Ellis. A Weak Convergence Approach to the Theory of Large Deviations. Wiley Series in Probability and Statistics, 1997

work page 1997

[19] [19]

Durrett and J

R. Durrett and J. Mayberry. Travelling waves of selective sweeps. Annals of Applied Probability, 21(2), 699–744, 2011

work page 2011

[20] [20]

Esser and A

M. Esser and A. Kraut. A general multi-scale description of metastable adaptive motion across fitness valleys. J. Math. Biol., 89(46), 2024

work page 2024

[21] [21]

Ethier and Thomas G, Kurtz

Stewart N. Ethier and Thomas G, Kurtz. Markov processes: characterization and convergence. John Wiley & Sons, 1986

work page 1986

[22] [22]

L. C. Evans Partial differential equations Graduate Studies in Mathematics Vol. 19, American Mathematical Society, 1998

work page 1998

[23] [23]

L. C. Evans and P. E. Souganidis. A PDE approach to geometric optics for certain semilinear parabolic equations. Indiana Univ. Math. J., 38(1):141–172, 1989

work page 1989

[24] [24]

Fang and O

M. Fang and O. Zeitouni. Slowdown for Time Inhomogeneous Branching Brownian Motion. J. Statistical Physics, 149:1-9, (2012)

work page 2012

[25] [25]

A. Fathi. Weak Kam Theorem in Lagrangian Dynamics. Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2016

work page 2016

[26] [26]

Limit theorems for large deviations and reaction-diffusion equations

M Freidlin. Limit theorems for large deviations and reaction-diffusion equations. The Annals of Probability, 13(3):639–675, 1985

work page 1985

[27] [27]

Forien, J

R. Forien, J. Garnier and F. Patout. Ancestral lineages in mutation-selection equilibria with moving optimum. Bull Math Biol., 84(93), 2022

work page 2022

[28] [28]

Fournier and S

N. Fournier and S. M´ el´ eard. A Microscopic Probabilistic Description of a Locally Regulated Population and Macroscopic Approximations. Ann. Appl. Probab., 14(4):1880-1919, 2004

work page 1919

[29] [29]

Hardy and S.C

R. Hardy and S.C. Harris. A Spine Approach to Branching Diffusions with Applications to Lp-Convergence of Martingales. S´ eminairede Probabilit´ esXLII, Lecture Notes in Mathematics, 1979:281-330, Springer Nature, 2009

work page 1979

[30] [30]

Henry, S

B. Henry, S. M´ el´ eard and V.C. Tran. Time reversal of spinal processes for linear and non-linear branching processes near stationarity. Electronic Journal of Probability, 28(32):1–27, 2023. 43

work page 2023

[31] [31]

Ikeda and S

N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes, North-Holland Pub- lishing Company, 1989

work page 1989

[32] [32]

P.-E. Jabin. Small populations corrections for selection-mutation models. Netw. Heterog. Media, 7(4), 805–836, 2012

work page 2012

[33] [33]

A. Jeddi. Asymptotic behavior of some stochastic models in population dynamics: a Hamilton-Jacobi approach. ArXiV arXiv:2602.20825, 2026

work page arXiv 2026

[34] [34]

Last and M

G. Last and M. Penrose. Lectures on the Poisson Process. IMS Textbook by Cambridge University Press (2017)

work page 2017

[35] [35]

A. Lorz, S. Mirrahimi and B Perthame. Dirac mass dynamics in a multidimensional nonlocal parabolic equation. Communications in Partial Differential Equations, 36, 1071–1098 (2011)

work page 2011

[36] [36]

Maillard and G

P. Maillard and G. Raoul and J. Tourniaire. Spreading speed of locally regulated population models in macroscopically heterogeneous environments (2024). ArXiv:2105.06985

work page arXiv 2024

[37] [37]

B. Mallein. Maximal displacement of a branching random walk in time- inhomogeneous environment. Stochastic Processes and Their Applications, 125(10), 3958–4019 (2015)

work page 2015

[38] [38]

A. Marguet. Uniform sampling in a structured branching population. Bernoulli, 25, 4A, 2649-2695, 2019

work page 2019

[39] [39]

M´ el´ eard and V.C

S. M´ el´ eard and V.C. Tran. Nonlinear historical superprocess approximations for population models with past dependence. Electronic Journal of Probability, 17(47):1-32, 2012

work page 2012

[40] [40]

Mirrahimi, G

S. Mirrahimi, G. Barles, B. Perthame, P.E. Souganidis. A singular Hamilton-Jacobi equation modeling the tail problem. SIAM J. Math. Anal., 44 (6), 4297–4319, 2012

work page 2012

[41] [41]

Perthame and G

B. Perthame and G. Barles. Dirac concentrations in Lotka-Volterra parabolic PDEs. Indiana Univ. Math. J. 57, 3275–3301 (2008)

work page 2008

[42] [42]

Perthame and M

B. Perthame and M. Gauduchon. Survival thresholds and mortality rates in adaptive dynamics: conciliating deterministic and stochastic simulations. Math. Med. Biol. 27(3), 195–210 (2010)

work page 2010

[43] [43]

Protter.Stochastic integration and differential equations, second Edition

P.E. Protter.Stochastic integration and differential equations, second Edition. Stochastic Modelling and Applied Probability, 21, Springer, Berlin, 2004

work page 2004

[44] [44]

Desmarais, E

C. Desmarais, E. Schertzer, Z. Talygi´ as. K-Branching random walk with noisy selection: large population limits and phase transitions. arXiv:2509.26254

work page arXiv

[45] [45]

Waxman and S

D. Waxman and S. Gavrilets. 20 Questions on Adaptive Dynamics. Journal of Evolutionary Biology 18, 1139-1154 (2005)

work page 2005

[46] [46]

Zeitouni

O. Zeitouni. Branching random walks and Gaussian fields. Lecture notes, (2012). 44

work page 2012