This special volume of the ESAIM Journal, Mathematical Modelling and Numerical Analysis, contains a collection of articles on probabilistic interpretations of some classes of nonlinear integro-differential equations. The selected contributions deal with a wide range of topics in applied probability theory and stochastic analysis, with applications in a variety of scientific disciplines, including physics, biology, fluid mechanics, molecular chemistry, financial mathematics and bayesian statistics. In this preface, we provide a brief presentation of the main contributions presented in this special volume. We have also included an introduction to classic probabilistic methods and a presentation of the more recent particle methods, with a synthetic picture of their mathematical foundations and their range of applications.
Mots-clés : Fokker-Planck equations, Vlasov diffusion models, fluid-lagrangian-velocities model, Boltzmann collision models, interacting jump processes, adaptive biasing force model, molecular dynamics, ground state energies, hidden Markov chain problems, Feynman-Kac semigroups, Dirichlet problems with boundary conditions, Poisson Boltzmann equations, mean field stochastic particle models, stochastic analysis, functional contraction inequalities, uniform propagation of chaos properties w.r.t. the time parameter
@article{M2AN_2010__44_5_805_0,
author = {Del Moral, Pierre and Hadjiconstantinou, Nicolas G.},
title = {An introduction to probabilistic methods with applications},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {805--829},
publisher = {EDP-Sciences},
volume = {44},
number = {5},
year = {2010},
doi = {10.1051/m2an/2010043},
mrnumber = {2731394},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2010043/}
}
TY - JOUR AU - Del Moral, Pierre AU - Hadjiconstantinou, Nicolas G. TI - An introduction to probabilistic methods with applications JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2010 SP - 805 EP - 829 VL - 44 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2010043/ DO - 10.1051/m2an/2010043 LA - en ID - M2AN_2010__44_5_805_0 ER -
%0 Journal Article %A Del Moral, Pierre %A Hadjiconstantinou, Nicolas G. %T An introduction to probabilistic methods with applications %J ESAIM: Modélisation mathématique et analyse numérique %D 2010 %P 805-829 %V 44 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2010043/ %R 10.1051/m2an/2010043 %G en %F M2AN_2010__44_5_805_0
Del Moral, Pierre; Hadjiconstantinou, Nicolas G. An introduction to probabilistic methods with applications. ESAIM: Modélisation mathématique et analyse numérique, Special Issue on Probabilistic methods and their applications, Tome 44 (2010) no. 5, pp. 805-829. doi : 10.1051/m2an/2010043. http://www.numdam.org/articles/10.1051/m2an/2010043/
[1] and , Low-variance direct Monte Carlo simulations using importance weights. ESAIM: M2AN 44 (2010) 1069-1083. | Numdam | Zbl
[2] , Nonlinear filtering for observations on a random vector field along a random vector field along a random path. Application to atmospheric turbulent velocities. ESAIM: M2AN 44 (2010) 921-945. | Numdam
[3] , and , Computational fluctuating fluid dynamics. ESAIM: M2AN 44 (2010) 1085-1105. | Numdam
[4] , , , and , Stochastic Lagrangian method for downscaling problems in meteorology. ESAIM: M2AN 44 (2010) 885-920. | Numdam
[5] , and , Quantitative concentration inequalities for empirical measures on non compact spaces. Prob. Theor. Relat. Fields 137 (2007) 541-593. | Zbl
[6] , and , Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation. ESAIM: M2AN 44 (2010) 867-884. | Numdam | Zbl
[7] , and , Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics. ESAIM: M2AN 44 (2010) 997-1048. | Numdam | Zbl
[8] and , Probabilistic methods for semilinear PDEs. Application to finance. ESAIM: M2AN 44 (2010) 1107-1133. | Numdam
[9] , Feynman-Kac formulae. Genealogical and interacting particle approximations, Series: Probability and Applications. Springer, New York (2004). | Zbl
[10] and , On the stability of Measure Valued Processes with Applications to filtering. C. R. Acad. Sci. Paris, Sér. I 329 (1999) 429-434. | Zbl
[11] and , On the stability of interacting processes with applications to filtering and genetic algorithms. Ann. Inst. Henri Poincaré 37 (2001) 155-194. | Numdam | Zbl
[12] and , Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Non-Linear Filtering, in Séminaire de Probabilités XXXIV, J. Azéma, M. Emery, M. Ledoux and M. Yor Eds., Lecture Notes in Mathematics 1729, Springer-Verlag, Berlin (2000) 1-145. | Numdam | Zbl
[13] and , Asymptotic stability of non linear semigroup of Feynman-Kac type. Ann. Fac. Sci. Toulouse Math. 11 (2002) 135-175. | Zbl
[14] and , Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman-Kac semigroups. ESAIM: PS 7 (2003) 171-208. | Numdam | Zbl
[15] and , Concentration inequalities for mean field particle models. Ann. Appl. Probab. (to appear).
[16] , and , A backward particle interpretation of Feynman-Kac formulae. ESAIM: M2AN 44 (2010) 947-975. | Numdam
[17] and , Large Deviations Techniques and Applications. Jones and Barlett Publishers, Boston (1993). | Zbl
[18] , and , Diffusion Monte Carlo method: Numerical analysis in a simple case. ESAIM: M2AN 41 (2007) 189-213. | Numdam | Zbl
[19] and , Markov processes: characterization and convergence, Wiley Series Probability & Statistics. Wiley (1986). | Zbl
[20] , Functional integration and partial differential equations, Annals of Mathematics Studies 109. Princeton University Press (1985). | Zbl
[21] , and , Existence, uniqueness and convergence of a particle approximation for the adaptive biasing force. ESAIM: M2AN 44 (2010) 831-865. | Numdam | Zbl
[22] , On distributions of certain Wiener functionals. Trans. Amer. Math. Soc. 65 (1949) 1-13. | Zbl
[23] and , Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics. Springer (2004). | Zbl
[24] , and , Long-time convergence of an adaptive biasing force method. Nonlinearity 21 (2008) 1155-1181. | Zbl
[25] , and , Elliptic equations of higher stochastic order. ESAIM: M2AN 44 (2010) 1135-1153. | Numdam | Zbl
[26] , Logarithmic Sobolev inequalities for some nonlinear PDE's. Stochastic Process. Appl. 95 (2001) 109-132. | Zbl
[27] , Convergence to equilibrium for granular media equations and their Euler schemes. Ann. Appl. Probab. 13 (2003) 540-560. | Zbl
[28] and , Concentration inequalities for Euler schemes, in Monte Carlo and Quasi Monte Carlo Methods 2004, H. Niederreiter and D. Talay Eds., Springer (2005) 355-372. | Zbl
[29] and , Monte Carlo methods for calculating some physical properties of large molecules. SIAM J. Sci. Comput. 26 (2004) 339-357. | Zbl
[30] , Propagation of chaos for a class of non-linear parabolic equation, in Stochastic Differential Equations, Lecture Series in Differential Equations, Catholic Univ., Air Force Office Sci. Res., Arlington (1967) 41-57. | Zbl
[31] , Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models, in Probabilistic Models for Nonlinear Partial Differential Equations 1627, Lecture Notes in Mathematics, Springer, Berlin-Heidelberg (1996) 44-95. | Zbl
[32] and , Quantitative uniform in time chaos propagation for Boltzmann collision processes. arXiv:1001.2994v1 (2010).
[33] , and , Numerical study of the systematic error in Monte Carlo schemes for semiconductors. ESAIM: M2AN 44 (2010) 1049-1068. | Numdam | Zbl
[34] , Stochastic integration and differential equations, Stochastic Modelling and Applied Probability 21. Springer-Verlag, Berlin (2005). | Zbl
[35] and , Continuous martingales and Brownian motion. Springer-Verlag, New York (1991). | Zbl
[36] , On the control of an interacting particle approximation of Schrödinger ground states. SIAM J. Math. Anal. 38 (2006) 824-844. | Zbl
[37] , On a probabilistic interpretation of shape derivatives of Dirichlet groundstates with application to Fermion nodes. ESAIM: M2AN 44 (2010) 977-995. | Numdam
[38] , Topics in propagation of chaos, in Lecture Notes in Math 1464, Springer, Berlin (1991) 164-251. | Zbl
[39] , Approximation of invariant measures on nonlinear Hamiltonian and dissipative stochastic different equations, in Progress in Stochastic Structural Dynamics 152, L.M.A.-C.N.R.S. (1999) 139-169.
[40] , Stochastic differential equation corresponding to the spatially homogeneous Boltzmann equation of Maxwellian and non cut-off type. J. Fac. Sci. Univ. Tokyo, Sect. IA, Math. 34 (1987) 351-369. | Zbl
[41] and , Weak Convergence and Empirical Processes. Second edition, Springer (2000). | Zbl
Cité par Sources :
