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.
@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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {44},
year = {2010},
pages = {805-829},
doi = {10.1051/m2an/2010043},
mrnumber = {2731394},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2010__44_5_805_0}
}
Del Moral, Pierre; Hadjiconstantinou, Nicolas G. An introduction to probabilistic methods with applications. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 44 (2010) pp. 805-829. doi : 10.1051/m2an/2010043. http://gdmltest.u-ga.fr/item/M2AN_2010__44_5_805_0/
[1] and , Low-variance direct Monte Carlo simulations using importance weights. ESAIM: M2AN 44 (2010) 1069-1083. | Numdam | Zbl 1200.82051
[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 | Zbl pre05798938
[3] , and , Computational fluctuating fluid dynamics. ESAIM: M2AN 44 (2010) 1085-1105. | Numdam | Zbl pre05798944
[4] , , , and , Stochastic Lagrangian method for downscaling problems in meteorology. ESAIM: M2AN 44 (2010) 885-920. | Numdam | Zbl pre05798937
[5] , and , Quantitative concentration inequalities for empirical measures on non compact spaces. Prob. Theor. Relat. Fields 137 (2007) 541-593. | Zbl 1113.60093
[6] , and , Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation. ESAIM: M2AN 44 (2010) 867-884. | Numdam | Zbl 1201.82029
[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 1204.82020
[8] and , Probabilistic methods for semilinear PDEs. Application to finance. ESAIM: M2AN 44 (2010) 1107-1133. | Numdam | Zbl pre05798945
[9] , Feynman-Kac formulae. Genealogical and interacting particle approximations, Series: Probability and Applications. Springer, New York (2004). | Zbl 1130.60003
[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 0935.92001
[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 0990.60005
[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 0963.60040
[13] and , Asymptotic stability of non linear semigroup of Feynman-Kac type. Ann. Fac. Sci. Toulouse Math. 11 (2002) 135-175. | Zbl 1042.60046
[14] and , Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman-Kac semigroups. ESAIM: PS 7 (2003) 171-208. | Numdam | Zbl 1040.81009
[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 | Zbl pre05798939
[17] and , Large Deviations Techniques and Applications. Jones and Barlett Publishers, Boston (1993). | Zbl 1177.60035
[18] , and , Diffusion Monte Carlo method: Numerical analysis in a simple case. ESAIM: M2AN 41 (2007) 189-213. | Numdam | Zbl 1135.81379
[19] and , Markov processes: characterization and convergence, Wiley Series Probability & Statistics. Wiley (1986). | Zbl 1089.60005
[20] , Functional integration and partial differential equations, Annals of Mathematics Studies 109. Princeton University Press (1985). | Zbl 0568.60057
[21] , and , Existence, uniqueness and convergence of a particle approximation for the adaptive biasing force. ESAIM: M2AN 44 (2010) 831-865. | Numdam | Zbl 1201.65011
[22] , On distributions of certain Wiener functionals. Trans. Amer. Math. Soc. 65 (1949) 1-13. | Zbl 0032.03501
[23] and , Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics. Springer (2004). | Zbl 0638.60065
[24] , and , Long-time convergence of an adaptive biasing force method. Nonlinearity 21 (2008) 1155-1181. | Zbl 1146.35320
[25] , and , Elliptic equations of higher stochastic order. ESAIM: M2AN 44 (2010) 1135-1153. | Numdam | Zbl 1203.65020
[26] , Logarithmic Sobolev inequalities for some nonlinear PDE's. Stochastic Process. Appl. 95 (2001) 109-132. | Zbl 1059.60084
[27] , Convergence to equilibrium for granular media equations and their Euler schemes. Ann. Appl. Probab. 13 (2003) 540-560. | Zbl 1031.60085
[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 1097.65012
[29] and , Monte Carlo methods for calculating some physical properties of large molecules. SIAM J. Sci. Comput. 26 (2004) 339-357. | Zbl 1075.65003
[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 0181.44401
[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 0864.60077
[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 1198.82068
[34] , Stochastic integration and differential equations, Stochastic Modelling and Applied Probability 21. Springer-Verlag, Berlin (2005). | Zbl 0694.60047
[35] and , Continuous martingales and Brownian motion. Springer-Verlag, New York (1991). | Zbl 0804.60001
[36] , On the control of an interacting particle approximation of Schrödinger ground states. SIAM J. Math. Anal. 38 (2006) 824-844. | Zbl 1174.60045
[37] , On a probabilistic interpretation of shape derivatives of Dirichlet groundstates with application to Fermion nodes. ESAIM: M2AN 44 (2010) 977-995. | Numdam | Zbl pre05798940
[38] , Topics in propagation of chaos, in Lecture Notes in Math 1464, Springer, Berlin (1991) 164-251. | Zbl 0732.60114
[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 0639.60105
[41] and , Weak Convergence and Empirical Processes. Second edition, Springer (2000). | Zbl 0862.60002