General approximation method for the distribution of Markov processes conditioned not to be killed
ESAIM: Probability and Statistics, Tome 18 (2014) , pp. 441-467.

We consider a strong Markov process with killing and prove an approximation method for the distribution of the process conditioned not to be killed when it is observed. The method is based on a Fleming-Viot type particle system with rebirths, whose particles evolve as independent copies of the original strong Markov process and jump onto each others instead of being killed. Our only assumption is that the number of rebirths of the Fleming-Viot type system doesn't explode in finite time almost surely and that the survival probability of the original process remains positive in finite time. The approximation method generalizes previous results and comes with a speed of convergence. A criterion for the non-explosion of the number of rebirths is also provided for general systems of time and environment dependent diffusion particles. This includes, but is not limited to, the case of the Fleming-Viot type system of the approximation method. The proof of the non-explosion criterion uses an original non-attainability of (0,0) result for pair of non-negative semi-martingales with positive jumps.

DOI : https://doi.org/10.1051/ps/2013045
Classification : 82C22,  65C50,  60K35,  60J60
Mots clés : particle systems, conditional distributions
@article{PS_2014__18__441_0,
     author = {Villemonais, Denis},
     title = {General approximation method for the distribution of Markov processes conditioned not to be killed},
     journal = {ESAIM: Probability and Statistics},
     pages = {441--467},
     publisher = {EDP-Sciences},
     volume = {18},
     year = {2014},
     doi = {10.1051/ps/2013045},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps/2013045/}
}
Villemonais, Denis. General approximation method for the distribution of Markov processes conditioned not to be killed. ESAIM: Probability and Statistics, Tome 18 (2014) , pp. 441-467. doi : 10.1051/ps/2013045. http://www.numdam.org/articles/10.1051/ps/2013045/

[1] I. Ben-Ari and R.G. Pinsky, Ergodic behavior of diffusions with random jumps from the boundary. Stoch. Proc. Appl. 119 (2009) 864-881. | MR 2499861 | Zbl 1158.60036

[2] M. Bieniek, K. Burdzy and S. Finch, Non-extinction of a Fleming−Viot particle model. Probab. Theory Relat. Fields (2011) 1-40. | MR 2925576 | Zbl 1253.60089

[3] M. Bieniek, K. Burdzy and S. Pal, Extinction of Fleming-Viot-type particle systems with strong drift. Electron. J. Probab. 17 (2012) 1-15. | MR 2878790 | Zbl 1258.60031

[4] K. Burdzy, R. Holyst, D. Ingerman and P. March, Configurational transition in a fleming-viot-type model and probabilistic interpretation of laplacian eigenfunctions. J. Phys. A 29 (1996) 2633-2642. | Zbl 0901.60054

[5] K. Burdzy, R. Holyst and P. March, A Fleming−Viot particle representation of the Dirichlet Laplacian. Commun. Math. Phys. 214 (200) 679-703. | MR 1800866 | Zbl 0982.60078

[6] P. Del Moral and L. Miclo, Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman-Kac semigroups. ESAIM: PS 7 (2003) 171-208. | Numdam | MR 1956078 | Zbl 1040.81009

[7] F. Delarue, Hitting time of a corner for a reflected diffusion in the square. Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008) 946-961. | Numdam | MR 2453777 | Zbl 1180.60035

[8] M.C. Delfour and J.-P. Zolésio, Shapes and geometries, Analysis, differential calculus, and optimization. Vol. 4, Advances in Design and Control. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2001). | MR 1855817 | Zbl 1002.49029

[9] P.A. Ferrari and N. Marić, Quasi stationary distributions and Fleming−Viot processes in countable spaces. Electron. J. Probab. 12 (2007) 684-702. | MR 2318407 | Zbl 1127.60088

[10] A. Friedman, Nonattainability of a set by a diffusion process. Trans. Amer. Math. Soc. 197 (1974) 245-271. | MR 346903 | Zbl 0289.60032

[11] I. Grigorescu and M. Kang, Hydrodynamic limit for a Fleming−Viot type system. Stoch. Proc. Appl. 110 (2004) 111-143. | MR 2052139 | Zbl 1075.60124

[12] I. Grigorescu and M. Kang, Ergodic properties of multidimensional Brownian motion with rebirth. Electron. J. Probab. 12 (2007) 1299-1322. | EuDML 128214 | MR 2346513 | Zbl 1127.60073

[13] I. Grigorescu and M. Kang, Immortal particle for a catalytic branching process. Probab. Theory Relat. Fields (2011) 1-29. | MR 2925577 | Zbl 1251.60064

[14] M. Kolb and D. Steinsaltz, Quasilimiting behavior for one-dimensional diffusions with killing. Ann. Probab. 40 (2012) 162-212. | MR 2917771 | Zbl 1278.60121

[15] M. Kolb and A. Wübker, On the Spectral Gap of Brownian Motion with Jump Boundary. Electron. J. Probab. 16 1214-1237. | MR 2827456 | Zbl 1234.60083

[16] M. Kolb and A. Wübker, Spectral Analysis of Diffusions with Jump Boundary. J. Funct. Anal. 261 1992-2012. | MR 2822321 | Zbl 1229.60093

[17] A. Lambert, Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct. Electron. J. Probab. 12 (2007) 420-446. | EuDML 128217 | MR 2299923 | Zbl 1127.60082

[18] J.-U. Löbus, A stationary Fleming−Viot type Brownian particle system. Math. Z. 263 (2009) 541-581. | MR 2545857 | Zbl 1176.60084

[19] S. Méléard and D. Villemonais, Quasi-stationary distributions and population processes. Probab. Surveys 9 (2012) 340-410. | MR 2994898 | Zbl 1261.92056

[20] P. Pollett, Quasi-stationary distributions: a bibliography. http://www.maths.uq.edu.au/˜pkp/papers/qsds/qsds.pdf

[21] S. Ramasubramanian, Hitting of submanifolds by diffusions. Probab. Theory Relat. Fields 78 (1988) 149-163. | MR 940875 | Zbl 0628.60079

[22] D. Revuz and M. Yor, Continuous martingales and Brownian motion, vol. 293, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 3rd edition (1999). | MR 1725357 | Zbl 0917.60006

[23] M. Rousset, On the control of an interacting particle estimation of Schrödinger ground states. SIAM J. Math. Anal. 38 (2006) 824-844. | MR 2262944 | Zbl 1174.60045

[24] D. Villemonais, Interacting particle systems and Yaglom limit approximation of diffusions with unbounded drift. Electron. J. Probab. 16 (2011) 1663-1692. | MR 2835250 | Zbl 1244.82052

[25] W. Zhen and X. Hua, Multi-dimensional reflected backward stochastic differential equations and the comparison theorem. Acta Math. Sci. 30 (2010) 1819-1836. | MR 2778651 | Zbl 1240.60166