• Bernoulli
  • Volume 24, Number 2 (2018), 1010-1032.

Exponential mixing properties for time inhomogeneous diffusion processes with killing

Pierre Del Moral and Denis Villemonais

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We consider an elliptic and time-inhomogeneous diffusion process with time-periodic coefficients evolving in a bounded domain of $\mathbb{R}^{d}$ with a smooth boundary. The process is killed when it hits the boundary of the domain (hard killing) or after an exponential time (soft killing) associated with some bounded rate function. The branching particle interpretation of the non absorbed diffusion again behaves as a set of interacting particles evolving in an absorbing medium. Between absorption times, the particles evolve independently one from each other according to the diffusion evolution operator; when a particle is absorbed, another selected particle splits into two offsprings. This article is concerned with the stability properties of these non absorbed processes. Under some classical ellipticity properties on the diffusion process and some mild regularity properties of the hard obstacle boundaries, we prove an uniform exponential strong mixing property of the process conditioned to not be killed. We also provide uniform estimates w.r.t. the time horizon for the interacting particle interpretation of these non-absorbed processes, yielding what seems to be the first result of this type for this class of diffusion processes evolving in soft and hard obstacles, both in homogeneous and non-homogeneous time settings.

Article information

Bernoulli, Volume 24, Number 2 (2018), 1010-1032.

Received: June 2015
Revised: March 2016
First available in Project Euclid: 21 September 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

process with absorption time-inhomogeneous diffusion process uniform mixing property


Del Moral, Pierre; Villemonais, Denis. Exponential mixing properties for time inhomogeneous diffusion processes with killing. Bernoulli 24 (2018), no. 2, 1010--1032. doi:10.3150/16-BEJ845.

Export citation


  • [1] Burdzy, K., Holyst, R., Ingerman, D. and March, P. (1996). Configurational transition in a Fleming–Viot-type model and probabilistic interpretation of Laplacian eigenfunctions. J. Phys. A 29 2633–2642.
  • [2] Burdzy, K., Hołyst, R. and March, P. (2000). A Fleming–Viot particle representation of the Dirichlet Laplacian. Comm. Math. Phys. 214 679–703.
  • [3] Cattiaux, P., Collet, P., Lambert, A., Martínez, S., Méléard, S. and San Martín, J. (2009). Quasi-stationary distributions and diffusion models in population dynamics. Ann. Probab. 37 1926–1969.
  • [4] Cattiaux, P. and Méléard, S. (2010). Competitive or weak cooperative stochastic Lotka–Volterra systems conditioned on non-extinction. J. Math. Biol. 60 797–829.
  • [5] Champagnat, N. and Villemonais, D. (2016). Exponential convergence to quasi-stationary distribution and $Q$-process. Probab. Theory Related Fields 164 243–283.
  • [6] Delfour, M.C. and Zolésio, J.-P. (2011). Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, 2nd ed. Advances in Design and Control 22. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM).
  • [7] Del Moral, P. (2013). Mean Field Simulation for Monte Carlo Integration. Monographs on Statistics and Applied Probability 126. Boca Raton, FL: CRC Press.
  • [8] Del Moral, P. and Doucet, A. (2004). Particle motions in absorbing medium with hard and soft obstacles. Stoch. Anal. Appl. 22 1175–1207.
  • [9] Del Moral, P. and Guionnet, A. (2001). On the stability of interacting processes with applications to filtering and genetic algorithms. Ann. Inst. Henri Poincaré Probab. Stat. 37 155–194.
  • [10] Del Moral, P. and Miclo, L. (2000). Branching and interacting particle systems approximations of Feynman–Kac formulae with applications to non-linear filtering. In Séminaire de Probabilités, XXXIV. Lecture Notes in Math. 1729 1–145. Berlin: Springer.
  • [11] Del Moral, P. and Miclo, L. (2000). A Moran particle system approximation of Feynman–Kac formulae. Stochastic Process. Appl. 86 193–216.
  • [12] Del Moral, P. and Miclo, L. (2002). On the stability of nonlinear Feynman–Kac semigroups. Ann. Fac. Sci. Toulouse Math. (6) 11 135–175.
  • [13] Del Moral, P. and Miclo, L. (2003). Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman–Kac semigroups. ESAIM Probab. Stat. 7 171–208.
  • [14] Ethier, S.N. and Krone, S.M. (1995). Comparing Fleming–Viot and Dawson–Watanabe processes. Stochastic Process. Appl. 60 171–190.
  • [15] Ethier, S.N. and Kurtz, T.G. (1986). Markov Processes: Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. New York: Wiley.
  • [16] Ferrari, P.A. and Marić, N. (2007). Quasi stationary distributions and Fleming–Viot processes in countable spaces. Electron. J. Probab. 12 684–702.
  • [17] Fleming, W.H. and Viot, M. (1979). Some measure-valued Markov processes in population genetics theory. Indiana Univ. Math. J. 28 817–843.
  • [18] Gong, G.L., Qian, M.P. and Zhao, Z.X. (1988). Killed diffusions and their conditioning. Probab. Theory Related Fields 80 151–167.
  • [19] Graham, C. and Méléard, S. (1997). Stochastic particle approximations for generalized Boltzmann models and convergence estimates. Ann. Probab. 25 115–132.
  • [20] Grigorescu, I. and Kang, M. (2004). Hydrodynamic limit for a Fleming–Viot type system. Stochastic Process. Appl. 110 111–143.
  • [21] Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes. North-Holland Mathematical Library 24. Amsterdam–New York: North-Holland; Tokyo: Kodansha, Ltd.
  • [22] Knobloch, R. and Partzsch, L. (2010). Uniform conditional ergodicity and intrinsic ultracontractivity. Potential Anal. 33 107–136.
  • [23] Kolb, M. and Wübker, A. (2011). Spectral analysis of diffusions with jump boundary. J. Funct. Anal. 261 1992–2012.
  • [24] Lindvall, T. and Rogers, L.C.G. (1986). Coupling of multidimensional diffusions by reflection. Ann. Probab. 14 860–872.
  • [25] Littin C., J. (2012). Uniqueness of quasistationary distributions and discrete spectra when $\infty$ is an entrance boundary and 0 is singular. J. Appl. Probab. 49 719–730.
  • [26] Méléard, S. and Villemonais, D. (2012). Quasi-stationary distributions and population processes. Probab. Surv. 9 340–410.
  • [27] Miura, Y. (2014). Ultracontractivity for Markov semigroups and quasi-stationary distributions. Stoch. Anal. Appl. 32 591–601.
  • [28] Pinsky, R.G. (1985). On the convergence of diffusion processes conditioned to remain in a bounded region for large time to limiting positive recurrent diffusion processes. Ann. Probab. 13 363–378.
  • [29] Priola, E. and Wang, F.-Y. (2006). Gradient estimates for diffusion semigroups with singular coefficients. J. Funct. Anal. 236 244–264.
  • [30] Rousset, M. (2006). On the control of an interacting particle estimation of Schrödinger ground states. SIAM J. Math. Anal. 38 824–844 (electronic).
  • [31] van Doorn, E.A. and Pollett, P.K. (2013). Quasi-stationary distributions for discrete-state models. European J. Oper. Res. 230 1–14.
  • [32] Villemonais, D. (2011). Interacting particle systems and Yaglom limit approximation of diffusions with unbounded drift. Electron. J. Probab. 16 1663–1692.
  • [33] Villemonais, D. (2013). Uniform tightness for time-inhomogeneous particle systems and for conditional distributions of time-inhomogeneous diffusion processes. Markov Process. Related Fields 19 543–562.
  • [34] Villemonais, D. (2014). General approximation method for the distribution of Markov processes conditioned not to be killed. ESAIM Probab. Stat. 18 441–467.
  • [35] Wang, F.-Y. (2004). Gradient estimates of Dirichlet heat semigroups and application to isoperimetric inequalities. Ann. Probab. 32 424–440.