An introduction to probabilistic methods with applications
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 5, p. 805-829

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.

DOI : https://doi.org/10.1051/m2an/2010043
Classification:  65M75,  68Q87,  60H35,  35Q68,  37N10,  35Q35,  35Q20
Keywords: 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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {44},
     number = {5},
     year = {2010},
     pages = {805-829},
     doi = {10.1051/m2an/2010043},
     mrnumber = {2731394},
     language = {en},
     url = {http://www.numdam.org/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, Volume 44 (2010) no. 5, pp. 805-829. doi : 10.1051/m2an/2010043. http://www.numdam.org/item/M2AN_2010__44_5_805_0/

[1] H.A. Al-Mohssen and N.G. Hadjiconstantinou, Low-variance direct Monte Carlo simulations using importance weights. ESAIM: M2AN 44 (2010) 1069-1083. | Numdam | Zbl 1200.82051

[2] C. Baehr, 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] J.B. Bell, A.L. Garcia and S.H. Williams, Computational fluctuating fluid dynamics. ESAIM: M2AN 44 (2010) 1085-1105. | Numdam | Zbl pre05798944

[4] F. Bernardin, M. Bossy, C. Chauvin, F. Jabir and A. Rousseau, Stochastic Lagrangian method for downscaling problems in meteorology. ESAIM: M2AN 44 (2010) 885-920. | Numdam | Zbl pre05798937

[5] F. Bolley, A. Guillin and C. Villani, Quantitative concentration inequalities for empirical measures on non compact spaces. Prob. Theor. Relat. Fields 137 (2007) 541-593. | Zbl 1113.60093

[6] F. Bolley, A. Guillin and F. Malrieu, 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] N. Champagnat, M. Bossy and D. Talay, 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] D. Crisan and K. Manolarakis, Probabilistic methods for semilinear PDEs. Application to finance. ESAIM: M2AN 44 (2010) 1107-1133. | Numdam | Zbl pre05798945

[9] P. Del Moral, Feynman-Kac formulae. Genealogical and interacting particle approximations, Series: Probability and Applications. Springer, New York (2004). | Zbl 1130.60003

[10] P. Del Moral and A. Guionnet, 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] P. Del Moral and A. Guionnet, 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] P. Del Moral and L. Miclo, 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] P. Del Moral and L. Miclo, Asymptotic stability of non linear semigroup of Feynman-Kac type. Ann. Fac. Sci. Toulouse Math. 11 (2002) 135-175. | Zbl 1042.60046

[14] P. Del Moral and L. Miclo, Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman-Kac semigroups. ESAIM: PS 7 (2003) 171-208. | Numdam | Zbl 1040.81009

[15] P. Del Moral and E. Rio, Concentration inequalities for mean field particle models. Ann. Appl. Probab. (to appear).

[16] P. Del Moral, A. Doucet and S.S. Singh, A backward particle interpretation of Feynman-Kac formulae. ESAIM: M2AN 44 (2010) 947-975. | Numdam | Zbl pre05798939

[17] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications. Jones and Barlett Publishers, Boston (1993). | Zbl 1177.60035

[18] M. El Makrini, B. Jourdain and T. Lelièvre, Diffusion Monte Carlo method: Numerical analysis in a simple case. ESAIM: M2AN 41 (2007) 189-213. | Numdam | Zbl 1135.81379

[19] S.N. Ethier and T.G. Kurtz, Markov processes: characterization and convergence, Wiley Series Probability & Statistics. Wiley (1986). | Zbl 1089.60005

[20] M. Freidlin, Functional integration and partial differential equations, Annals of Mathematics Studies 109. Princeton University Press (1985). | Zbl 0568.60057

[21] B. Jourdain, R. Roux and T. Lelièvre, Existence, uniqueness and convergence of a particle approximation for the adaptive biasing force. ESAIM: M2AN 44 (2010) 831-865. | Numdam | Zbl 1201.65011

[22] M. Kac, On distributions of certain Wiener functionals. Trans. Amer. Math. Soc. 65 (1949) 1-13. | Zbl 0032.03501

[23] I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics. Springer (2004). | Zbl 0638.60065

[24] T. Lelièvre, M. Rousset and G. Stoltz, Long-time convergence of an adaptive biasing force method. Nonlinearity 21 (2008) 1155-1181. | Zbl 1146.35320

[25] S. Lototsky, B. Rozovsky and X. Wan, Elliptic equations of higher stochastic order. ESAIM: M2AN 44 (2010) 1135-1153. | Numdam | Zbl 1203.65020

[26] F. Malrieu, Logarithmic Sobolev inequalities for some nonlinear PDE's. Stochastic Process. Appl. 95 (2001) 109-132. | Zbl 1059.60084

[27] F. Malrieu, Convergence to equilibrium for granular media equations and their Euler schemes. Ann. Appl. Probab. 13 (2003) 540-560. | Zbl 1031.60085

[28] F. Malrieu and D. Talay, 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] M. Mascagni and N.A. Simonov, Monte Carlo methods for calculating some physical properties of large molecules. SIAM J. Sci. Comput. 26 (2004) 339-357. | Zbl 1075.65003

[30] H.P. Mckean, 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] S. Méléard, 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] S. Mischler and C. Mouhot, Quantitative uniform in time chaos propagation for Boltzmann collision processes. arXiv:1001.2994v1 (2010).

[33] O. Muscato, W. Wagner and V. Di Stefano, Numerical study of the systematic error in Monte Carlo schemes for semiconductors. ESAIM: M2AN 44 (2010) 1049-1068. | Numdam | Zbl 1198.82068

[34] P. Protter, Stochastic integration and differential equations, Stochastic Modelling and Applied Probability 21. Springer-Verlag, Berlin (2005). | Zbl 0694.60047

[35] D. Revuz and M. Yor, Continuous martingales and Brownian motion. Springer-Verlag, New York (1991). | Zbl 0804.60001

[36] M. Rousset, On the control of an interacting particle approximation of Schrödinger ground states. SIAM J. Math. Anal. 38 (2006) 824-844. | Zbl 1174.60045

[37] M. Rousset, 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] A.-S. Sznitman, Topics in propagation of chaos, in Lecture Notes in Math 1464, Springer, Berlin (1991) 164-251. | Zbl 0732.60114

[39] D. Talay, 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] H. Tanaka, 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] A.W. Van Der Vaart and J.A. Wellner, Weak Convergence and Empirical Processes. Second edition, Springer (2000). | Zbl 0862.60002