### Metadata

### Abstract

Growth-fragmentation processes describe the evolution of systems of cells which grow continuously and fragment suddenly; they are used in models of cell division and protein polymerisation. Typically, we may expect that in the long run, the concentrations of cells with given masses increase at some exponential rate, and that, after compensating for this, they arrive at an asymptotic profile. Up to now, this question has mainly been studied for the average behavior of the system, often by means of a natural partial integro-differential equation and the associated spectral theory. However, the behavior of the system as a whole, rather than only its average, is more delicate. In this work, we show that a criterion found by one of the authors for exponential ergodicity on average is actually sufficient to deduce stronger results about the convergence of the entire collection of cells to a certain asymptotic profile, and we find some improved explicit conditions for this to occur.

### References

[AH76] Strong limit theorems for general supercritical branching processes with applications to branching diffusions, Z. Wahrscheinlichkeitstheor. Verw. Geb., Volume 36 (1976) no. 3, pp. 195-212 | DOI | MR | Zbl

[BBCK18] Martingales in self-similar growth-fragmentations and their connections with random planar maps, Probab. Theory Relat. Fields, Volume 172 (2018) no. 3-4, pp. 663-724 | DOI | MR | Zbl

[BBH + 15] Growth rates of the population in a branching Brownian motion with an inhomogeneous breeding potential, Stochastic Processes Appl., Volume 125 (2015) no. 5, pp. 2096-2145 | DOI | MR | Zbl

[BCG13] Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates, Kinet. Relat. Models, Volume 6 (2013) no. 2, pp. 219-243 | DOI | MR | Zbl

[BCG20] Ergodic Behavior of Non-conservative Semigroups via Generalized Doeblin’s Conditions, Acta Appl. Math., Volume 166 (2020), pp. 29-72 | DOI | MR | Zbl

[BCGM19] A non-conservative Harris’ ergodic theorem (2019) (https://arxiv.org/abs/1903.03946v1)

[BDMT11] Limit theorems for Markov processes indexed by continuous time Galton–Watson trees, Ann. Appl. Probab., Volume 21 (2011) no. 6, pp. 2263-2314 | DOI | MR | Zbl

[Ber03] Multifractal spectra of fragmentation processes, J. Stat. Phys., Volume 113 (2003) no. 3-4, pp. 411-430 | DOI | MR | Zbl

[Ber06] Random fragmentation and coagulation processes, Cambridge Studies in Advanced Mathematics, Volume 102, Cambridge University Press, 2006, viii+280 pages | DOI | MR | Zbl

[Ber17] Markovian growth-fragmentation processes, Bernoulli, Volume 23 (2017) no. 2, pp. 1082-1101 | DOI | MR | Zbl

[Ber19] On a Feynman–Kac approach to growth-fragmentation semigroups and their asymptotic behaviors, J. Funct. Anal., Volume 277 (2019) no. 11, 108270, p. 29 | DOI | MR | Zbl

[BG20] Asynchronous exponential growth of the growth-fragmentation equation with unbounded fragmentation rate, J. Evol. Equ., Volume 20 (2020), pp. 375-401 | DOI | MR | Zbl

[Big92] Uniform convergence of martingales in the branching random walk, Ann. Probab., Volume 20 (1992) no. 1, pp. 137-151 | DOI | MR | Zbl

[BK04] Measure change in multitype branching, Adv. Appl. Probab., Volume 36 (2004) no. 2, pp. 544-581 | DOI | MR | Zbl

[BR05] Discretization methods for homogeneous fragmentations, J. London Math. Soc. (2), Volume 72 (2005) no. 1, pp. 91-109 | DOI | MR | Zbl

[BW18] A probabilistic approach to spectral analysis of growth-fragmentation equations, J. Funct. Anal., Volume 274 (2018) no. 8, pp. 2163-2204 | DOI | MR | Zbl

[Cav20] On a family of critical growth-fragmentation semigroups and refracted Lévy processes, Acta Appl. Math., Volume 166 (2020) no. 1, pp. 161-186 | DOI | MR | Zbl

[CDP18] Long-Time Asymptotics for Polymerization Models, Commun. Math. Phys., Volume 363 (2018) no. 1, pp. 111-137 | DOI | MR | Zbl

[Cha91] Product martingales and stopping lines for branching Brownian motion, Ann. Probab., Volume 19 (1991) no. 3, pp. 1195-1205 | DOI | MR | Zbl

[CHHK19] Multi-species Neutron Transport Equation, J. Stat. Phys., Volume 176 (2019) no. 2, pp. 425-455 | DOI | MR | Zbl

[Clo17] Limit theorems for some branching measure-valued processes, Adv. Appl. Probab., Volume 49 (2017) no. 2, pp. 549-580 | DOI | MR | Zbl

[CRY17] Law of large numbers for branching symmetric Hunt processes with measure-valued branching rates, J. Theor. Probab., Volume 30 (2017) no. 3, pp. 898-931 | DOI | MR | Zbl

[CS07] Limit theorems for branching Markov processes, J. Funct. Anal., Volume 250 (2007) no. 2, pp. 374-399 | DOI | MR | Zbl

[CV16] Exponential convergence to quasi-stationary distribution and $Q$-process, Probab. Theory Relat. Fields, Volume 164 (2016) no. 1-2, pp. 243-283 | DOI | MR | Zbl

[CV20] Practical criteria for R-positive recurrence of unbounded semigroups, Electron. Commun. Probab., Volume 25 (2020), 6 | DOI | MR | Zbl

[Dad17] Asymptotics of self-similar growth-fragmentation processes, Electron. J. Probab., Volume 22 (2017), 27, p. 30 | DOI | MR | Zbl

[DB92] The supercritical Galton–Watson process in varying environments, Stoch. Proc. Appl., Volume 42 (1992) no. 1, pp. 39-47 | DOI | MR | Zbl

[DDGW18] Relative entropy method for measure solutions of the growth-fragmentation equation, SIAM J. Math. Anal., Volume 50 (2018) no. 6, pp. 5811-5824 | DOI | MR | Zbl

[DJG10] Eigenelements of a general aggregation-fragmentation model, Math. Models Methods Appl. Sci., Volume 20 (2010) no. 5, pp. 757-783 | DOI | MR | Zbl

[DM04] Feynman–Kac formulae. Genealogical and interacting particle systems with applications, Probability and its Applications, Springer, 2004, xviii+555 pages | DOI | Zbl

[EHK10] Strong law of large numbers for branching diffusions, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 46 (2010) no. 1, pp. 279-298 | DOI | Numdam | MR | Zbl

[EN00] One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, Volume 194, Springer, 2000, xxii+586 pages | MR | Zbl

[Esc20] On the non existence of non negative solutions to a critical Growth-Fragmentation Equation, Ann. Fac. Sci. Toulouse, Math., Volume 29 (2020) no. 1, pp. 177-220 | DOI | Zbl

[GHH07] Exponential growth rates in a typed branching diffusion, Ann. Appl. Probab., Volume 17 (2007) no. 2, pp. 609-653 | DOI | MR | Zbl

[HH09] A spine approach to branching diffusions with applications to ${L}^{p}$-convergence of martingales, Séminaire de Probabilités XLII (Catherine, Donati-Martin; al, eds.) (Lecture Notes in Mathematics) Volume 1979, Springer, 2009, pp. 281-330 | DOI | MR | Zbl

[HHK16] Branching Brownian motion in a strip: survival near criticality, Ann. Probab., Volume 44 (2016) no. 1, pp. 235-275 | DOI | MR | Zbl

[HHK19] Stochastic Methods for the Neutron Transport Equation II: Almost sure growth (2019) (https://arxiv.org/abs/1901.00220, to appear in The Annals of Applied Probability)

[HKV18] Stochastic Methods for the Neutron Transport Equation I: Linear Semigroup asymptotics (2018) (https://arxiv.org/abs/1810.01779, to appear in The Annals of Applied Probability)

[Jag89] General branching processes as Markov fields, Stoch. Proc. Appl., Volume 32 (1989) no. 2, pp. 183-212 | DOI | MR | Zbl

[JN84] The growth and composition of branching populations, Adv. Appl. Probab., Volume 16 (1984) no. 2, pp. 221-259 | DOI | MR | Zbl

[KP76] Sur certaines martingales de Benoit Mandelbrot, Adv. Math., Volume 22 (1976) no. 2, pp. 131-145 | DOI | MR | Zbl

[Mar19] A law of large numbers for branching Markov processes by the ergodicity of ancestral lineages, ESAIM Probab. Stat., Volume 23 (2019), pp. 638-661 | DOI | MR

[MS16] Spectral analysis of semigroups and growth-fragmentation equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 33 (2016) no. 3, pp. 849-898 | DOI | MR | Zbl

[Ner81] On the convergence of supercritical general (C-M-J) branching processes, Z. Wahrscheinlichkeitstheor. Verw. Geb., Volume 57 (1981) no. 3, pp. 365-395 | DOI | MR | Zbl

[Per07] Transport equations in biology, Frontiers in Mathematics, Birkhäuser, 2007 | Zbl

[Pey74] Turbulence et dimension de Hausdorff, C. R. Acad. Sci., Paris, Sér. A, Volume 278 (1974), pp. 567-569 | MR | Zbl

[Sav69] The explosion problem for branching Markov process, Osaka J. Math., Volume 6 (1969) no. 2, pp. 375-395 | MR | Zbl

[Shi08] Exponential growth of the numbers of particles for branching symmetric $\alpha $-stable processes, J. Math. Soc. Japan, Volume 60 (2008) no. 1, pp. 75-116 | DOI | MR | Zbl

[Shi15] Branching random walks. École d’Été de Probabilités de Saint-Flour XLII – 2012, Lecture Notes in Mathematics, Volume 2151, Springer, 2015, x+133 pages | DOI | Zbl

[Shi20] A growth-fragmentation model related to Ornstein–Uhlenbeck type processes, Ann. Inst. H. Poincaré Probab. Statist., Volume 56 (2020) no. 1, pp. 580-611 | DOI | MR | Zbl