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Existence, uniqueness and ergodicity for the centered Fleming–Viot process

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  • Champagnat, Nicolas
  • Hass, Vincent

Abstract

Motivated by questions of ergodicity for shift invariant Fleming–Viot process, we consider the centered Fleming–Viot process Ztt⩾0 defined by Zt≔τ−id,Yt♯Yt, where Ytt⩾0 is the original Fleming–Viot process. Our goal is to characterize the centered Fleming–Viot process with a martingale problem. To establish the existence of a solution to this martingale problem, we exploit the original Fleming–Viot martingale problem and asymptotic expansions. The proof of uniqueness is based on a weakened version of the duality method, allowing us to prove uniqueness for initial conditions admitting finite moments. We also provide counter examples showing that our approach based on the duality method cannot be expected to give uniqueness for more general initial conditions. Finally, we establish ergodicity properties with exponential convergence in total variation for the centered Fleming–Viot process and characterize the invariant measure.

Suggested Citation

  • Champagnat, Nicolas & Hass, Vincent, 2023. "Existence, uniqueness and ergodicity for the centered Fleming–Viot process," Stochastic Processes and their Applications, Elsevier, vol. 166(C).
    • Handle: RePEc:eee:spapps:v:166:y:2023:i:c:s0304414923001837
    DOI: 10.1016/j.spa.2023.09.006
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    References listed on IDEAS

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    1. U. Dieckmann & R. Law, 1996. "The Dynamical Theory of Coevolution: A Derivation from Stochastic Ecological Processes," Working Papers wp96001, International Institute for Applied Systems Analysis.
    2. Ethier, S. N. & Kurtz, Thomas G., 1994. "Convergence to Fleming-Viot processes in the weak atomic topology," Stochastic Processes and their Applications, Elsevier, vol. 54(1), pages 1-27, November.
    3. Champagnat, Nicolas, 2006. "A microscopic interpretation for adaptive dynamics trait substitution sequence models," Stochastic Processes and their Applications, Elsevier, vol. 116(8), pages 1127-1160, August.
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