M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series. For sale by the Superintendent of Documents, U.S. Government Printing Office, vol.55, 1964.

L. Addario-berry and B. Reed, Minima in branching random walks, The Annals of Probability, vol.37, issue.3, pp.1044-1079, 2009.
DOI : 10.1214/08-AOP428

URL : https://hal.archives-ouvertes.fr/hal-00795281

E. Aïdékon, Convergence in law of the minimum of a branching random walk, The Annals of Probability, vol.41, issue.3A, pp.1362-1426, 2013.
DOI : 10.1214/12-AOP750

E. Aïdékon and B. Jaffuel, Survival of branching random walks with absorption. Stochastic Process, Appl, vol.121, issue.9, pp.1901-1937, 2011.

E. Aïdékon and Z. Shi, Weak convergence for the minimal position in a branching random walk: A simple proof, Periodica Mathematica Hungarica, vol.143, issue.17, pp.43-54, 2010.
DOI : 10.1007/s10998-010-3043-x

J. D. Biggins, The first-and last-birth problems for a multitype age-dependent branching process Advances in Appl, Probability, vol.8, issue.3, pp.446-459, 1976.

J. D. Biggins, Branching out, Probability and mathematical genetics, pp.113-134, 2010.
DOI : 10.1017/CBO9781139107174.007

J. D. Biggins and A. E. Kyprianou, Measure change in multitype branching, Advances in Applied Probability, vol.64, issue.02, pp.544-581, 2004.
DOI : 10.1214/aop/1024404291

A. Bovier and I. Kurkova, Much ado about Derrida's GREM. In Spin glasses, Lecture Notes in Math, pp.81-115, 1900.

H. Brezis, Analyse fonctionnelle Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master's Degree], Théorie et applications. [Theory and applications], 1983.

D. A. Darling, On the Supremum of a Certain Gaussian Process, The Annals of Probability, vol.11, issue.3, pp.803-806, 1983.
DOI : 10.1214/aop/1176993527

M. Fang, Tightness for Maxima of Generalized Branching Random Walks, Journal of Applied Probability, vol.49, issue.03, pp.652-670, 2012.
DOI : 10.1214/08-AOP428

M. Fang and O. Zeitouni, Consistent Minimal Displacement of Branching Random Walks, Electronic Communications in Probability, vol.15, issue.0, pp.106-118, 2010.
DOI : 10.1214/ECP.v15-1533

M. Fang and O. Zeitouni, Branching random walks in time inhomogeneous environments, Electronic Journal of Probability, vol.17, issue.0, 2012.
DOI : 10.1214/EJP.v17-2253

M. Fang and O. Zeitouni, Slowdown for Time Inhomogeneous Branching Brownian Motion, Journal of Statistical Physics, vol.125, issue.1, pp.1-9, 2012.
DOI : 10.1007/s10955-012-0581-z

G. Faraud, Y. Hu, and Z. Shi, Almost sure convergence for stochastically biased random walks on trees. Probab. Theory Related Fields, pp.3-4621, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00767694

A. F. Filippov, Differential equations with discontinuous righthand sides, volume 18 of Mathematics and its Applications (Soviet Series), 1988.

K. Fleischmann and V. Wachtel, Lower deviation probabilities for supercritical Galton???Watson processes???, Annales de l'Institut Henri Poincare (B) Probability and Statistics, vol.43, issue.2, pp.233-255, 2007.
DOI : 10.1016/j.anihpb.2006.03.001

N. Gantert, Y. Hu, and Z. Shi, Asymptotics for the survival probability in a killed branching random walk, Annales de l'Institut Henri Poincar??, Probabilit??s et Statistiques, vol.47, issue.1, pp.111-129, 2011.
DOI : 10.1214/10-AIHP362

URL : https://hal.archives-ouvertes.fr/hal-00579979

J. M. Hammersley, Postulates for Subadditive Processes, The Annals of Probability, vol.2, issue.4, pp.652-680, 1974.
DOI : 10.1214/aop/1176996611

C. Simon, M. I. Harris, and . Roberts, The many-to-few lemma and multiple spines, 2015.

P. Hartman, Ordinary differential equations Corrected reprint of the second (1982) edition, Classics in Applied Mathematics . Society for Industrial and Applied Mathematics (SIAM), vol.38, p.658490, 2002.

Y. Hu and Z. Shi, Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees, The Annals of Probability, vol.37, issue.2, pp.742-789, 2009.
DOI : 10.1214/08-AOP419

URL : https://hal.archives-ouvertes.fr/hal-00414685

K. Itô, H. P. Mckean, and J. , Diffusion processes and their sample paths. Die Grundlehren der Mathematischen Wissenschaften, Band 125, 1965.

K. Itô, H. P. Mckean, and J. , Diffusion processes and their sample paths, 1974.

J. Kahane and J. Peyrière, Sur certaines martingales de Benoit Mandelbrot, Advances in Mathematics, vol.22, issue.2, pp.131-145, 1976.
DOI : 10.1016/0001-8708(76)90151-1

I. Karatzas and S. E. Shreve, Brownian motion and stochastic calculus, Graduate Texts in Mathematics, vol.113, 1991.

J. F. Kingman, The First Birth Problem for an Age-dependent Branching Process, The Annals of Probability, vol.3, issue.5, pp.790-801, 1975.
DOI : 10.1214/aop/1176996266

S. Kurcyusz, On the existence and nonexistence of Lagrange multipliers in Banach spaces, Journal of Optimization Theory and Applications, vol.17, issue.1, pp.81-110, 1976.
DOI : 10.1007/BF00933349

G. Louchard, The brownian excursion area: a numerical analysis, Computers & Mathematics with Applications, vol.10, issue.6, pp.413-417, 1984.
DOI : 10.1016/0898-1221(84)90071-3

P. Maillard and O. Zeitouni, Slowdown in branching Brownian motion with inhomogeneous variance, Annales de l'Institut Henri Poincar??, Probabilit??s et Statistiques, vol.52, issue.3, 2015.
DOI : 10.1214/15-AIHP675

B. Mallein, Position of the rightmost individual in a branching random walk through a series of interfaces, 2015.

J. Nolen, J. Roquejoffre, and L. Ryzhik, Power-Like Delay in Time Inhomogeneous Fisher-KPP Equations, Communications in Partial Differential Equations, vol.18, issue.3, pp.475-505, 2015.
DOI : 10.1214/12-AOP753

J. Peyrière, Turbulence et dimension de Hausdorff, C. R. Acad. Sci. Paris Sér. A, vol.278, pp.567-569, 1974.

A. I. Sakhanenko, Rate of convergence in the invariance principle for variables with exponential moments that are not identically distributed. In Limit theorems for sums of random variables, Trudy Inst. Mat. Nauka " Sibirsk. Otdel, vol.3, pp.4-49, 1984.

L. Takács, Random Walk Processes and their Applications in Order Statistics, The Annals of Applied Probability, vol.2, issue.2, pp.435-459, 1992.
DOI : 10.1214/aoap/1177005710

O. Vallée and M. Soares, Airy functions and applications to physics, 2004.

A. Zettl, Sturm-Liouville theory, volume 121 of Mathematical Surveys and Monographs, 2005.