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An introduction to probabilistic methods with applications

Published online by Cambridge University Press:  26 August 2010

Pierre Del Moral
Affiliation:
Centre INRIA Bordeaux et Sud-Ouest & Institut de Mathématiques de Bordeaux, Université de Bordeaux I, 351 cours de la Libération, 33405 Talence Cedex, France. Pierre.Del-Moral@inria.fr
Nicolas G. Hadjiconstantinou
Affiliation:
Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge MA 02139-4307, USA. ngh@mit.edu
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Abstract

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.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2010

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References

Al-Mohssen, H.A. and Hadjiconstantinou, N.G., Low-variance direct Monte Carlo simulations using importance weights. ESAIM: M2AN 44 (2010) 10691083. CrossRef
Baehr, C., 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) 921945. CrossRef
Bell, J.B., Garcia, A.L. and Williams, S.H., Computational fluctuating fluid dynamics. ESAIM: M2AN 44 (2010) 10851105. CrossRef
Bernardin, F., Bossy, M., Chauvin, C., Jabir, F. and Rousseau, A., Stochastic Lagrangian method for downscaling problems in meteorology. ESAIM: M2AN 44 (2010) 885920. CrossRef
Bolley, F., Guillin, A. and Villani, C., Quantitative concentration inequalities for empirical measures on non compact spaces. Prob. Theor. Relat. Fields 137 (2007) 541593. CrossRef
Bolley, F., Guillin, A. and Malrieu, F., Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation. ESAIM: M2AN 44 (2010) 867884. CrossRef
Champagnat, N., Bossy, M. and Talay, D., Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics. ESAIM: M2AN 44 (2010) 9971048.
Crisan, D. and Manolarakis, K., Probabilistic methods for semilinear PDEs. Application to finance. ESAIM: M2AN 44 (2010) 11071133. CrossRef
P. Del Moral, Feynman-Kac formulae. Genealogical and interacting particle approximations, Series: Probability and Applications. Springer, New York (2004).
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.
Del Moral, P. and Guionnet, A., On the stability of interacting processes with applications to filtering and genetic algorithms. Ann. Inst. Henri Poincaré 37 (2001) 155194. CrossRef
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.
Del Moral, P. and Miclo, L., Asymptotic stability of non linear semigroup of Feynman-Kac type. Ann. Fac. Sci. Toulouse Math. 11 (2002) 135175. CrossRef
Del Moral, P. and Miclo, L., Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman-Kac semigroups. ESAIM: PS 7 (2003) 171208.
P. Del Moral and E. Rio, Concentration inequalities for mean field particle models. Ann. Appl. Probab. (to appear).
Del Moral, P., Doucet, A. and Singh, S.S., A backward particle interpretation of Feynman-Kac formulae. ESAIM: M2AN 44 (2010) 947975. CrossRef
A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications. Jones and Barlett Publishers, Boston (1993).
El Makrini, M., Jourdain, B. and Lelièvre, T., Diffusion Monte Carlo method: Numerical analysis in a simple case. ESAIM: M2AN 41 (2007) 189213. CrossRef
S.N. Ethier and T.G. Kurtz, Markov processes: characterization and convergence, Wiley Series Probability & Statistics. Wiley (1986).
M. Freidlin, Functional integration and partial differential equations, Annals of Mathematics Studies 109. Princeton University Press (1985).
Jourdain, B., Roux, R. and Lelièvre, T., Existence, uniqueness and convergence of a particle approximation for the adaptive biasing force. ESAIM: M2AN 44 (2010) 831865. CrossRef
Kac, M., On distributions of certain Wiener functionals. Trans. Amer. Math. Soc. 65 (1949) 113. CrossRef
I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics. Springer (2004).
Lelièvre, T., Rousset, M. and Stoltz, G., Long-time convergence of an adaptive biasing force method. Nonlinearity 21 (2008) 11551181. CrossRef
Lototsky, S., Rozovsky, B. and Wan, X., Elliptic equations of higher stochastic order. ESAIM: M2AN 44 (2010) 11351153. CrossRef
Malrieu, F., Logarithmic Sobolev inequalities for some nonlinear PDE's. Stochastic Process. Appl. 95 (2001) 109132. CrossRef
Malrieu, F., Convergence to equilibrium for granular media equations and their Euler schemes. Ann. Appl. Probab. 13 (2003) 540560. CrossRef
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.
Mascagni, M. and Simonov, N.A., Monte Carlo methods for calculating some physical properties of large molecules. SIAM J. Sci. Comput. 26 (2004) 339357. CrossRef
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.
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.
S. Mischler and C. Mouhot, Quantitative uniform in time chaos propagation for Boltzmann collision processes. arXiv:1001.2994v1 (2010).
Muscato, O., Wagner, W. and Di Stefano, V., Numerical study of the systematic error in Monte Carlo schemes for semiconductors. ESAIM: M2AN 44 (2010) 10491068. CrossRef
P. Protter, Stochastic integration and differential equations, Stochastic Modelling and Applied Probability 21. Springer-Verlag, Berlin (2005).
D. Revuz and M. Yor, Continuous martingales and Brownian motion. Springer-Verlag, New York (1991).
M. Rousset, On the control of an interacting particle approximation of Schrödinger ground states. SIAM J. Math. Anal. 38 (2006) 824–844.
Rousset, M., On a probabilistic interpretation of shape derivatives of Dirichlet groundstates with application to Fermion nodes. ESAIM: M2AN 44 (2010) 977995. CrossRef
A.-S. Sznitman, Topics in propagation of chaos, in Lecture Notes in Math 1464, Springer, Berlin (1991) 164–251.
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.
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.
A.W. van der Vaart and J.A. Wellner, Weak Convergence and Empirical Processes. Second edition, Springer (2000).