]. E. Arjas, E. Nummelin, and R. L. Tweedie, Semi-Markov processes on a general state space: ??-theory and quasi-stationarity, Journal of the Australian Mathematical Society, vol.22, issue.02, pp.187-20081, 1980.
DOI : 10.1093/qmath/25.1.485

K. B. Athreya and P. E. Ney, Branching processes, Die Grundlehren der mathematischen Wissenschaften, 0196.

R. Azaï-s, J. Bardet, A. Génadot, N. Krell, and P. Zitt, Piecewise deterministic Markov process ??? recent results, Journées MAS 2012, pp.276-290, 2014.
DOI : 10.1051/proc/201444017

M. Baudel and N. Berglund, Spectral Theory for Random Poincar?? Maps, SIAM Journal on Mathematical Analysis, vol.49, issue.6, pp.4319-4375, 2017.
DOI : 10.1137/16M1103816
URL : http://arxiv.org/pdf/1611.04869

N. Berglund and D. Landon, Mixed-mode oscillations and interspike interval statistics in the stochastic FitzHugh???Nagumo model, Nonlinearity, vol.25, issue.8, pp.2303-2335, 2012.
DOI : 10.1088/0951-7715/25/8/2303
URL : https://hal.archives-ouvertes.fr/hal-00591089

G. Birkhoff, Extensions of Jentzsch's theorem, Trans. Amer. Math. Soc, vol.85, pp.219-227, 1957.

F. M. Buchmann, Simulation of stopped diffusions, Journal of Computational Physics, vol.202, issue.2, pp.446-462, 2005.
DOI : 10.1016/j.jcp.2004.07.009

D. L. Burkholder, Martingale Transforms, The Annals of Mathematical Statistics, vol.37, issue.6, pp.1494-1504, 1966.
DOI : 10.1214/aoms/1177699141

P. Cattiaux, P. Collet, A. Lambert, S. Martínez, S. Méléard et al., Quasi-stationary distributions and diffusion models in population dynamics, The Annals of Probability, vol.37, issue.5, pp.1926-1969, 2009.
DOI : 10.1214/09-AOP451
URL : https://hal.archives-ouvertes.fr/hal-00138521

P. Cattiaux and S. Méléard, Competitive or weak cooperative stochastic Lotka???Volterra systems conditioned on non-extinction, Journal of Mathematical Biology, vol.65, issue.4, pp.797-829, 2010.
DOI : 10.5802/afst.1105
URL : https://hal.archives-ouvertes.fr/hal-00336156/document

J. A. Cavender, Quasi-stationary distributions of birth-and-death processes, Advances in Applied Probability, vol.2, issue.03, pp.570-586, 1978.
DOI : 10.2307/3212128

N. Champagnat, K. Coulibaly-pasquier, and D. Villemonais, Exponential convergence to quasi-stationary distribution for multi-dimensional diffusion processes. ArXiv e-prints, 2016.

N. Champagnat, P. Diaconis, and L. Miclo, On Dirichlet eigenvectors for neutral two-dimensional Markov chains, Electronic Journal of Probability, vol.17, issue.0, p.41, 2012.
DOI : 10.1214/EJP.v17-1830
URL : https://hal.archives-ouvertes.fr/hal-00672938

N. Champagnat and S. Roelly, Limit theorems for conditioned multitype Dawson-Watanabe processes and Feller diffusions, Electronic Journal of Probability, vol.13, issue.0, pp.777-810, 2008.
DOI : 10.1214/EJP.v13-504
URL : https://hal.archives-ouvertes.fr/inria-00164758

N. Champagnat and D. Villemonais, Exponential convergence to quasistationary distribution and Q-process. Probab. Theory Related Fields, pp.243-283, 2016.
DOI : 10.1007/s00440-014-0611-7
URL : https://hal.archives-ouvertes.fr/hal-00973509

N. Champagnat and D. Villemonais, Population processes with unbounded extinction rate conditioned to non-extinction. ArXiv e-prints, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01395731

N. Champagnat and D. Villemonais, Exponential convergence to quasistationary distribution for absorbed one-dimensional diffusions with killing. ALEA Lat, Am. J. Probab. Math. Stat, vol.14, issue.1, pp.177-199, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01217843

N. Champagnat and D. Villemonais, Lyapunov criteria for uniform convergence of conditional distributions of absorbed Markov processes. ArXiv e-prints, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01503697

N. Champagnat and D. Villemonais, Uniform convergence of conditional distributions for absorbed one-dimensional diffusions, J. Appl. Probab, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01166960

N. Champagnat and D. Villemonais, Uniform convergence of penalized time-inhomogeneous Markov processes, ESAIM: Probability and Statistics, 2018.
DOI : 10.1051/ps/2017022
URL : https://hal.archives-ouvertes.fr/hal-01290222

S. D. Chatterji, An $L^p$-Convergence Theorem, The Annals of Mathematical Statistics, vol.40, issue.3, pp.1068-1070, 1969.
DOI : 10.1214/aoms/1177697609
URL : http://doi.org/10.1214/aoms/1177697609

J. Chazottes, P. Collet, and S. Méléard, Sharp asymptotics for the quasistationary distribution of birth-and-death processes. Probab. Theory Related Fields, pp.285-332, 2016.

J. Chazottes, P. Collet, and S. Méléard, On time scales and quasistationary distributions for multitype birth-and-death processes. ArXiv eprints, 2017.

P. Collet, S. Martínez, and J. San-martín, Asymptotic Laws for One-Dimensional Diffusions Conditioned to Nonabsorption, The Annals of Probability, vol.23, issue.3, pp.1300-1314, 1995.
DOI : 10.1214/aop/1176988185
URL : http://doi.org/10.1214/aop/1176988185

P. Collet, S. Martínez, and J. San-martín, Quasi-stationary distributions. Probability and its Applications, Markov chains, diffusions and dynamical systems
URL : https://hal.archives-ouvertes.fr/hal-00138521

C. Coron, Stochastic modeling and eco-evolution of a diploid population, 2013.
URL : https://hal.archives-ouvertes.fr/pastel-00918257

C. Coron, S. Méléard, E. Porcher, and A. Robert, Quantifying the Mutational Meltdown in Diploid Populations, The American Naturalist, vol.181, issue.5, pp.623-636, 2013.
DOI : 10.1086/670022

J. N. Darroch and E. Seneta, On Quasi-Stationary distributions in absorbing discrete-time finite Markov chains, Journal of Applied Probability, vol.84, issue.01, pp.88-100, 1965.
DOI : 10.1214/aoms/1177705139

J. N. Darroch and E. Seneta, On quasi-stationary distributions in absorbing continuous-time finite Markov chains, Journal of Applied Probability, vol.28, issue.01, pp.192-196, 1967.
DOI : 10.1111/j.1467-842X.1966.tb00168.x

E. B. Davies and B. Simon, Ultracontractivity and the heat kernel for Schr??dinger operators and Dirichlet Laplacians, Journal of Functional Analysis, vol.59, issue.2, pp.335-395, 1984.
DOI : 10.1016/0022-1236(84)90076-4

P. and D. Moral, Feynman-Kac formulae. Probability and its Applications, Genealogical and interacting particle systems with applications, 2004.
URL : https://hal.archives-ouvertes.fr/hal-00426415

P. and D. Moral, Mean field simulation for Monte Carlo integration, of Monographs on Statistics and Applied Probability
URL : https://hal.archives-ouvertes.fr/hal-00932211

P. , D. Moral, and D. Villemonais, Exponential mixing properties for time inhomogeneous diffusion processes with killing, Bernoulli, vol.24, issue.2, pp.1010-1032, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01593882

M. Faure and S. J. Schreiber, Quasi-stationary distributions for randomly perturbed dynamical systems, The Annals of Applied Probability, vol.24, issue.2, pp.553-598, 2014.
DOI : 10.1214/13-AAP923
URL : https://hal.archives-ouvertes.fr/hal-01474257

P. A. Ferrari, H. Kesten, and S. Martínez, $R$-positivity, quasi-stationary distributions and ratio limit theorems for a class of probabilistic automata, The Annals of Applied Probability, vol.6, issue.2
DOI : 10.1214/aoap/1034968146

P. A. Ferrari, H. Kesten, S. Martínez, and P. Picco, Existence of Quasi-Stationary Distributions. A Renewal Dynamical Approach, The Annals of Probability, vol.23, issue.2, pp.501-521, 1995.
DOI : 10.1214/aop/1176988277

P. A. Ferrari, S. Martínez, and P. Picco, Some Properties of Quasi Stationary Distributions in the Birth and Death Chains: A Dynamical Approach, Instabilities and nonequilibrium structures, III (Valparaíso, pp.177-187, 1989.
DOI : 10.1007/978-94-011-3442-2_16

P. A. Ferrari, S. Martínez, and P. Picco, Existence of non-trivial quasi-stationary distributions in the birth-death chain, Advances in Applied Probability, vol.56, issue.04, pp.795-813, 1992.
DOI : 10.2307/3214486

A. Friedman, Partial differential equations of parabolic type, N.J, 1964.

E. Gobet, Weak approximation of killed diffusion using Euler schemes. Stochastic Process, Appl, vol.87, issue.2, pp.167-197, 2000.

E. Gobet, Euler schemes and half-space approximation for the simulation of diffusion in a domain, ESAIM: Probability and Statistics, vol.8, issue.4, pp.261-297, 2001.
DOI : 10.1016/0304-4149(94)90118-X

G. L. Gong, M. P. Qian, and Z. X. Zhao, Killed diffusions and their conditioning . Probab. Theory Related Fields, pp.151-167, 1988.
DOI : 10.1007/bf00348757

P. Good, The limiting behavior of transient birth and death processes conditioned on survival, Journal of the Australian Mathematical Society, vol.3, issue.04, pp.716-722, 1968.
DOI : 10.2307/1993021

F. Gosselin, Aysmptotic Behavior of Absorbing Markov Chains Conditional on Nonabsorption for Applications in Conservation Biology, The Annals of Applied Probability, vol.11, issue.1, pp.261-284, 2001.
DOI : 10.1214/aoap/998926993

T. E. Harris, The theory of branching processes. Die Grundlehren der Mathematischen Wissenschaften, Bd, N.J, vol.119, 1963.

C. R. Heathcote, E. Seneta, D. Vere, and -. , A Refinement of Two Theorems in the Theory of Branching Processes, Theory of Probability & Its Applications, vol.12, issue.2, pp.341-346, 1967.
DOI : 10.1137/1112033

A. Hening and M. Kolb, Quasistationary distributions for one-dimensional diffusions with singular boundary points. ArXiv e-prints, 2014.

G. Hinrichs, M. Kolb, and V. Wachtel, Persistence of one-dimensional AR(1)-sequences. ArXiv e-prints, 2018.

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

A. Joffe and F. Spitzer, On multitype branching processes with ?? ??? 1, Journal of Mathematical Analysis and Applications, vol.19, issue.3, pp.409-430, 1967.
DOI : 10.1016/0022-247X(67)90001-7
URL : https://doi.org/10.1016/0022-247x(67)90001-7

O. Kallenberg, Foundations of modern probability. Probability and its Applications, 2002.

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

S. Karlin and J. Mcgregor, The classification of birth and death processes, Transactions of the American Mathematical Society, vol.86, issue.2, pp.366-400, 1957.
DOI : 10.1090/S0002-9947-1957-0094854-8

M. Kijima and E. Seneta, Some results for quasi-stationary distributions of birth-death processes, Journal of Applied Probability, vol.24, issue.03, pp.503-511, 1991.
DOI : 10.1016/0021-9045(87)90038-4

J. F. Kingman, The Exponential Decay of Markov Transition Probabilities, Proc. London Math. Soc. (3), pp.337-358, 1963.
DOI : 10.1112/plms/s3-13.1.337

R. Knobloch and L. Partzsch, Uniform Conditional Ergodicity and Intrinsic Ultracontractivity, Potential Analysis, vol.23, issue.2, pp.107-136, 2010.
DOI : 10.1007/978-3-642-65970-6

M. Kolb and D. Steinsaltz, Quasilimiting behavior for one-dimensional diffusions with killing, The Annals of Probability, vol.40, issue.1, pp.162-212, 2012.
DOI : 10.1214/10-AOP623

N. V. Krylov and M. V. Safonov, A CERTAIN PROPERTY OF SOLUTIONS OF PARABOLIC EQUATIONS WITH MEASURABLE COEFFICIENTS, Mathematics of the USSR-Izvestiya, vol.16, issue.1, pp.161-175, 1980.
DOI : 10.1070/IM1981v016n01ABEH001283

A. Lambert, Quasi-Stationary Distributions and the Continuous-State Branching Process Conditioned to Be Never Extinct, Electronic Journal of Probability, vol.12, issue.0, pp.420-446, 2007.
DOI : 10.1214/EJP.v12-402

J. Littin and C. , Uniqueness of Quasistationary Distributions and Discrete Spectra when ??? is an Entrance Boundary and 0 is Singular, Journal of Applied Probability, vol.49, issue.03, pp.719-730, 2012.
DOI : 10.1214/09-AOP451

M. Lladser and J. San-martín, Domain of attraction of the quasi-stationary distributions for the Ornstein-Uhlenbeck process, Journal of Applied Probability, vol.II, issue.02, pp.511-520, 2000.
DOI : 10.1214/aoap/1034968146

P. Maillard, The $\lambda$-invariant measures of subcritical Bienaym?????Galton???Watson processes, Bernoulli, vol.24, issue.1, pp.297-315, 2018.
DOI : 10.3150/16-BEJ877

R. Mannella, Absorbing boundaries and optimal stopping in a stochastic differential equation, Physics Letters A, vol.254, issue.5, pp.257-262, 1999.
DOI : 10.1016/S0375-9601(99)00117-6

S. Martínez and J. San-martín, Classification of killed one-dimensional diffusions, The Annals of Probability, vol.32, issue.1A, pp.530-552, 2004.
DOI : 10.1214/aop/1078415844

S. Martínez, J. San-martín, and D. Villemonais, Existence and Uniqueness of a Quasistationary Distribution for Markov Processes with Fast Return from Infinity, Journal of Applied Probability, vol.12, issue.03, pp.756-768, 2014.
DOI : 10.1214/aop/1176988277

S. Méléard and D. Villemonais, Quasi-stationary distributions and population processes, Probability Surveys, vol.9, issue.0, pp.340-410, 2012.
DOI : 10.1214/11-PS191

S. Meyn and R. L. Tweedie, Markov chains and stochastic stability, 2009.

S. P. Meyn and R. L. Tweedie, Stability of Markovian processes. III. Foster- Lyapunov criteria for continuous-time processes, Adv. in Appl. Probab, vol.25, issue.3, pp.518-548, 1993.

Y. Miura, Ultracontractivity for Markov Semigroups and Quasi-Stationary Distributions, Stochastic Analysis and Applications, vol.50, issue.4, pp.591-601, 2014.
DOI : 10.1016/j.jfa.2007.05.003

Y. Ogura, Asymptotic behavior of multitype Galton-Watson processes, Journal of Mathematics of Kyoto University, vol.15, issue.2, pp.251-302, 1975.
DOI : 10.1215/kjm/1250523066

R. G. Pinsky, On the Convergence of Diffusion Processes Conditioned to Remain in a Bounded Region for Large Time to Limiting Positive Recurrent Diffusion Processes, The Annals of Probability, vol.13, issue.2, pp.363-378, 1985.
DOI : 10.1214/aop/1176992996

R. G. Pinsky, Explicit and almost explicit spectral calculations for diffusion operators, Journal of Functional Analysis, vol.256, issue.10, pp.3279-3312, 2009.
DOI : 10.1016/j.jfa.2008.08.012
URL : https://doi.org/10.1016/j.jfa.2008.08.012

P. E. Protter, Stochastic integration and differential equations, Applications of Mathematics, vol.21, 2004.

B. Roynette, P. Vallois, and M. Yor, Some penalisations of the Wiener measure, Japanese Journal of Mathematics, vol.79, issue.1, pp.263-290, 2006.
DOI : 10.1007/s11537-006-0507-0
URL : https://hal.archives-ouvertes.fr/hal-00128461

E. Seneta, D. Vere, and -. , On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states, Journal of Applied Probability, vol.1, issue.02, pp.403-434, 1966.
DOI : 10.1093/qmath/13.1.7

D. Steinsaltz and S. N. Evans, Markov mortality models: implications of quasistationarity and varying initial distributions, Theoretical Population Biology, vol.65, issue.4, pp.319-337, 2004.
DOI : 10.1016/j.tpb.2003.10.007

D. W. Stroock and S. R. Varadhan, Multidimensional diffusion processes, Classics in Mathematics, 2006.
DOI : 10.1007/3-540-28999-2

E. A. Van-doorn, Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes, Advances in Applied Probability, vol.53, issue.04, pp.683-700, 1991.
DOI : 10.1111/j.1467-842X.1969.tb00300.x

E. A. Van-doorn, Conditions for the existence of quasi-stationary distributions for birth-death processes with killing. Stochastic Process, Appl, vol.122, issue.6, pp.2400-2410, 2012.

E. A. Van-doorn and P. K. Pollett, Quasi-stationary distributions for discrete-state models, European Journal of Operational Research, vol.230, issue.1, pp.1-14, 2013.
DOI : 10.1016/j.ejor.2013.01.032

A. Velleret, Non-uniform exponential convergence to qsd, with a possibly stabilizing extinction, preparation, 2018.

D. Vere and -. , Ergodic properties of nonnegative matrices. I, Pacific Journal of Mathematics, vol.22, issue.2, pp.361-386, 1967.
DOI : 10.2140/pjm.1967.22.361

D. Villemonais, Minimal quasi-stationary distribution approximation for a birth and death process, Electronic Journal of Probability, vol.20, issue.0, 2015.
DOI : 10.1214/EJP.v20-3482
URL : https://hal.archives-ouvertes.fr/hal-00983773

J. Wang, First Eigenvalue of One-dimensional Diffusion Processes, Electronic Communications in Probability, vol.14, issue.0, pp.232-244, 2009.
DOI : 10.1214/ECP.v14-1464

J. Wang, Sharp bounds for the first eigenvalue of symmetric Markov processes and their applications, Acta Mathematica Sinica, English Series, vol.349, issue.10, p.28, 1995.
DOI : 10.1007/978-3-642-32298-3

A. M. Yaglom, Certain limit theorems of the theory of branching random processes. Doklady Akad, Nauk SSSR (N.S.), vol.56, pp.795-798, 1947.