Criteria for Exponential Convergence to Quasi-Stationary Distributions and Applications to Multi-Dimensional Diffusions

  • Nicolas Champagnat
  • Koléhè Abdoulaye Coulibaly-Pasquier
  • Denis Villemonais
Part of the Lecture Notes in Mathematics book series (LNM, volume 2215)


We consider general Markov processes with absorption and provide criteria ensuring the exponential convergence in total variation of the distribution of the process conditioned not to be absorbed. The first one is based on two-sided estimates on the transition kernel of the process and the second one on gradient estimates on its semigroup. We apply these criteria to multi-dimensional diffusion processes in bounded domains of \(\mathbb {R}^d\) or in compact Riemannian manifolds with boundary, with absorption at the boundary.


Markov processes Diffusions in Riemannian manifolds Diffusions in bounded domains Absorption at the boundary Quasi-stationary distributions Q-process Uniform exponential mixing Two-sided estimates Gradient estimates 

2010 Mathematics Subject Classification

Primary: 60J60 37A25 60B10 60F99 Secondary: 60J75 60J70 


  1. 1.
    D.G. Aronson, Non-negative solutions of linear parabolic equations. Ann. Scuola Norm. Sup. Pisa (3) 22, 607–694 (1968)Google Scholar
  2. 2.
    K. Bogdan, T. Grzywny, M. Ryznar, Heat kernel estimates for the fractional Laplacian with Dirichlet conditions. Ann. Probab. 38(5), 1901–1923 (2010)MathSciNetCrossRefGoogle Scholar
  3. 3.
    P. Cattiaux, S. Méléard, Competitive or weak cooperative stochastic lotka-volterra systems conditioned to non-extinction. J. Math. Biol. 60(6), 797–829 (2010)MathSciNetCrossRefGoogle Scholar
  4. 4.
    P. Cattiaux, P. Collet, A. Lambert, S. Martínez, S. Méléard, J.S. Martín, Quasi-stationary distributions and diffusion models in population dynamics. Ann. Probab. 37(5), 1926–1969 (2009)MathSciNetCrossRefGoogle Scholar
  5. 5.
    N. Champagnat, D. Villemonais, Uniform convergence of conditional distributions for absorbed one-dimensional diffusions (2015). ArXiv e-printsGoogle Scholar
  6. 6.
    N. Champagnat, D. Villemonais, Exponential convergence to quasi-stationary distribution and Q-process. Probab. Theory Relat. Fields 164(1–2), 243–283 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    N. Champagnat, D. Villemonais, Uniform convergence to the Q-process. Electron. Commun. Probab. 22, 1–7 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Z.-Q. Chen, P. Kim, R. Song, Heat kernel estimates for Δ + Δ α∕2 in C 1, 1 open sets. J. Lond. Math. Soc. (2) 84(1), 58–80 (2011)Google Scholar
  9. 9.
    Z.-Q. Chen, P. Kim, R. Song, Dirichlet heat kernel estimates for fractional Laplacian with gradient perturbation. Ann. Probab. 40(6), 2483–2538 (2012)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Z.-Q. Chen, P. Kim, R. Song, Stability of Dirichlet heat kernel estimates for non-local operators under Feynman-Kac perturbation. Trans. Am. Math. Soc. 367(7), 5237–5270 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    E.B. Davies, B. Simon, Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians. J. Funct. Anal. 59(2), 335–395 (1984)MathSciNetCrossRefGoogle Scholar
  12. 12.
    P. Del Moral, D. Villemonais, Exponential mixing properties for time inhomogeneous diffusion processes with killing. Bernoulli 24(2), 1010–1032 (2018)MathSciNetCrossRefGoogle Scholar
  13. 13.
    E.B. Dynkin, Diffusions, Superdiffusions and Partial Differential Equations. American Mathematical Society Colloquium Publications, vol. 50 (American Mathematical Society, Providence, 2002)Google Scholar
  14. 14.
    A. Hening, M. Kolb, Quasistationary distributions for one-dimensional diffusions with singular boundary points (2014). ArXiv e-printsGoogle Scholar
  15. 15.
    O. Kallenberg, Foundations of Modern Probability. Probability and Its Applications (New York), 2nd edn. (Springer, New York, 2002)Google Scholar
  16. 16.
    K.-Y. Kim, P. Kim, Two-sided estimates for the transition densities of symmetric Markov processes dominated by stable-like processes in C 1, η open sets. Stoch. Process. Appl. 124(9), 3055–3083 (2014)MathSciNetCrossRefGoogle Scholar
  17. 17.
    P. Kim, R. Song, Estimates on Green functions and Schrödinger-type equations for non-symmetric diffusions with measure-valued drifts. J. Math. Anal. Appl. 332(1), 57–80 (2007)MathSciNetCrossRefGoogle Scholar
  18. 18.
    R. Knobloch, L. Partzsch, Uniform conditional ergodicity and intrinsic ultracontractivity. Potential Anal. 33, 107–136 (2010)MathSciNetCrossRefGoogle Scholar
  19. 19.
    M. Kolb, A. Wübker, Spectral analysis of diffusions with jump boundary. J. Funct. Anal. 261(7), 1992–2012 (2011)MathSciNetCrossRefGoogle Scholar
  20. 20.
    J.-F. Le Gall, Brownian Motion, Martingales, and Stochastic Calculus. Graduate Texts in Mathematics, vol. 274, French edn. (Springer, Berlin, 2016)Google Scholar
  21. 21.
    J. Lierl, L. Saloff-Coste, The Dirichlet heat kernel in inner uniform domains: local results, compact domains and non-symmetric forms. J. Funct. Anal. 266(7), 4189–4235 (2014)MathSciNetCrossRefGoogle Scholar
  22. 22.
    T. Lindvall, L.C.G. Rogers, Coupling of multidimensional diffusions by reflection. Ann. Probab. 14(3), 860–872 (1986)MathSciNetCrossRefGoogle Scholar
  23. 23.
    J.C. Littin, Uniqueness of quasistationary distributions and discrete spectra when is an entrance boundary and 0 is singular. J. Appl. Probab. 49(3), 719–730 (2012)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Y. Miura, Ultracontractivity for Markov semigroups and quasi-stationary distributions. Stoch. Anal. Appl. 32(4), 591–601 (2014)MathSciNetCrossRefGoogle Scholar
  25. 25.
    D. Nualart, The Malliavin Calculus and Related Topics. Probability and Its Applications (New York), 2nd edn. (Springer, Berlin, 2006)Google Scholar
  26. 26.
    E. Priola, F.-Y. Wang, Gradient estimates for diffusion semigroups with singular coefficients. J. Funct. Anal. 236(1), 244–264 (2006)MathSciNetCrossRefGoogle Scholar
  27. 27.
    D.W. Stroock, S.R.S. Varadhan, Multidimensional Diffusion Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 233 (Springer, Berlin, 1979)Google Scholar
  28. 28.
    F.-Y. Wang, Sharp explicit lower bounds of heat kernels. Ann. Probab. 25(4), 1995–2006 (1997)MathSciNetCrossRefGoogle Scholar
  29. 29.
    F.-Y. Wang, Gradient estimates of Dirichlet heat semigroups and application to isoperimetric inequalities. Ann. Probab. 32(1A), 424–440 (2004)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Q.S. Zhang, Gaussian bounds for the fundamental solutions of ∇(Au) + Bu − u t = 0. Manuscr. Math. 93(3), 381–390 (1997)CrossRefGoogle Scholar
  31. 31.
    Q.S. Zhang, The boundary behavior of heat kernels of Dirichlet Laplacians. J. Differ. Equ. 182(2), 416–430 (2002)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Nicolas Champagnat
    • 1
    • 2
    • 3
  • Koléhè Abdoulaye Coulibaly-Pasquier
    • 1
    • 2
    • 3
  • Denis Villemonais
    • 1
    • 2
    • 3
  1. 1.Université de Lorraine, IECL, UMR 7502Vandœuvre-lès-NancyFrance
  2. 2.CNRS, IECL, UMR 7502Vandœuvre-lès-NancyFrance
  3. 3.InriaVillers-lès-NancyFrance

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