On the stability of nonlinear Feynman-Kac semigroups

Del Moral, Pierre; Miclo, Laurent

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 11 (2002) no. 2, pp. 135-175.

[1] Bakry (D.). - L'hypercontractivité et son utilisation en théorie des semi-groupes, École d'été de St. Flour XXII-1992, Lecture Notes in Math, 1581, Ed. P. Bernard. | MR | Zbl

[2] Carlen (E.A.). - Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous maxwellian gas, Comm. Math. Phys. 199 (1999), n° 3, pp. 521-546. | MR | Zbl

[3] Carlen (E.A.), Carvalho (M.C.) & Wennberg (B.). - Entropic convergence for solutions of the Boltzmann equation with general physical initial data, Transport Theory Statist. Phys. 26 (1997), n° 3, pp. 373-378. | Zbl

[4] Clark (J.M.C.).- The design of robust approximation to stochastic differential equations of nonlinear filtering, Comm. Systems and Random Process Theory, (Ed. J.K. Skwirzinski), Sithof and Noordhoof (1978). | MR

[5] Davis (E.B.). - Heat Kernels and Spectral Theory, Cambridge University Press, 1989. | MR | Zbl

[6] Del Moral (P.) & Guionnet (A.). - On the stability of interacting processes with applications to filtering and genetic algorithms, Ann. de l'Inst. H. Poincaré Probab. Statist., to appear (2001). | Numdam | Zbl

[7] Del Moral (P.), Jacod (J.) & Protter (Ph.). - The Monte-Carlo Method for filtering with discrete time observations, Probability Theory and Related Fields, to appear (2001). | Zbl

[8] Del Moral (P.) & Miclo (L.). - Branching and interacting particle systems approximations of Feynman-Kac formulae with applications to non-linear filtering, Séminaire de Probabilités XXXIV, Lecture Notes in Mathematics, Springer, Vol. 1729 (2000), pp. 1-145. | Numdam | Zbl

[9] Del Moral (P.) & Miclo (L.). - About the strong propagation of chaos for interacting particle approximations of Feynman-Kac formulae, Preprint (2000).

[10] Dobrushin (R.L.). - Central limit theorem for nonstationnary Markov chains, I,II, Theory of Proba and its applications, Vol 1, number 1 and 4 (1956), pp. 66-80 and pp. 330-385. | MR | Zbl

[11] Ethier (S.) & Kurtz (T.). - Markov Processes, Characterization and Convergence, Wiley series in probability and mathematical statistics, John Wiley and Sons, New York, 1986. | Zbl

[12] Fleming (W.H.) & Mitter (S.K.). - Optimal control and nonlinear filtering for nondegenerate diffusion processes, Stochastics, Vol. 8 (1972), pp. 63-77. | Zbl

[13] Jacod (J.) & Shiryaev (A.N.). - Limit Theorems for Stochastic Processes, Springer- Verlag, 288 (1987). | Zbl

[14] Kushner (H.). - A robust discrete state approximation to the optimal nonlinear filter for a diffusion, Stochastics, Vol. 3 (1979), pp. 75-83. | MR | Zbl

[15] Méléard (S.). - Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models, In D. Talay and L. Tubaro, editors, Probabilistic Models for Nonlinear Partial Differential Equations, Montecatini Terme, 1995, Lecture Notes in Mathematics, 1627, Springer-Verlag, 1996. | MR | Zbl

[16] Pardoux (E.). - Équation du lissage non linéaire, de la prédiction et du lissage, Stochastics, vol. 6 (1982), pp. 193-231. | MR | Zbl

[17] Revuz (D.) & Yor (M.). - Continuous Martingales and Brownian Motion, Springer-Verlag, 293, 2nd Ed. (1991). | Zbl

[18] Rozovskiiĭ (B.). - Stochastic partial differential equations that arise in nonlinear filtering problem, U.M.N. XXVII, 3 (1972), pp. 213-214. | Zbl

[19] Tamura (Y.). - Free energy and the convergence of distributions of diffusions of McKean type, J. Fac. Sci. Univ. Tokyo, Sect. IA, Math., 34, pp. 443-484. | MR | Zbl