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Benjamin Jourdain

Bio: Benjamin Jourdain is an academic researcher from University of Paris. The author has contributed to research in topics: Stochastic differential equation & Nonlinear system. The author has an hindex of 26, co-authored 166 publications receiving 2226 citations. Previous affiliations of Benjamin Jourdain include École des ponts ParisTech & French Institute for Research in Computer Science and Automation.


Papers
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Journal ArticleDOI
TL;DR: In this paper, a non-linear micro-macro model of polymeric fluids in the case of a shear flow was analyzed and the existence of a unique solution to the stochastic differential equation which rules the evolution of a representative polymer in the flow was proved.

156 citations

Posted Content
18 Jul 2007
TL;DR: In this paper, the authors studied general nonlinear stochastic differential equa- tions, where the usual Brownian motion is replaced by a Levy process, and they proved that the time-marginals of the solutions are abso- lutely continuous with respect to the Lebesgue measure.
Abstract: In this paper we study general nonlinear stochastic differential equa- tions, where the usual Brownian motion is replaced by a Levy process. Moreover, we do not suppose that the coefficient multiplying the increments of this process is linear in the time-marginals of the solution as is the case in the classical McKean- Vlasov model. We first study existence, uniqueness and particle approximations for these stochastic differential equations. When the driving process is a pure jump Levy process with a smooth but unbounded Levy measure, we develop a stochastic calculus of variations to prove that the time-marginals of the solutions are abso- lutely continuous with respect to the Lebesgue measure. In the case of a symmetric stable driving process, we deduce the existence of a function solution to a nonlinear integro-differential equation involving the fractional Laplacian.

109 citations

Journal ArticleDOI
TL;DR: In this article, a stochastic differential equation which is nonlinear in the sense that both the diffusion and the drift coefficients depend locally on the density of the time marginal of the solution was studied.
Abstract: In this paper, we are interested in a stochastic differential equation which is nonlinear in the following sense: both the diffusion and the drift coefficients depend locally on the density of the time marginal of the solution. When the law of the initial data has a smooth density with respect to Lebesgue measure, we prove existence and uniqueness for this equation. Under more restrictive assumptions on the density, we approximate the solution by a system of n moderately interacting diffusion processes and obtain a trajectorial propagation of chaos result. Finally, we study the fluctuations associated with the convergence of the empirical measure of the system to the law of the solution of the nonlinear equation. In this situation, the convergence rate is different from √ n .

106 citations

Journal ArticleDOI
TL;DR: In this article, the long-time behavior of some micro-macro models for polymeric fluids (Hookean model and FENE model) in various settings (shear flow, general bounded domain with homogeneous Dirichlet boundary conditions on the velocity and non-homogeneous Diriclet boundary condition on the velocities) was investigated.
Abstract: In this paper, we investigate the long-time behavior of some micro-macro models for polymeric fluids (Hookean model and FENE model), in various settings (shear flow, general bounded domain with homogeneous Dirichlet boundary conditions on the velocity, general bounded domain with non-homogeneous Dirichlet boundary conditions on the velocity). We use both probabilistic approaches (coupling methods) and analytic approaches (entropy methods).

96 citations

Journal ArticleDOI
TL;DR: In this paper, a stochastic algorithm based mainly on Monte Carlo Methods and Applications 5(1) (1999) 1; Stochastic particle approximations for Smoluchowski's coagulation equation.

80 citations


Cited by
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Journal ArticleDOI
TL;DR: Convergence of Probability Measures as mentioned in this paper is a well-known convergence of probability measures. But it does not consider the relationship between probability measures and the probability distribution of probabilities.
Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

5,689 citations

Book ChapterDOI
01 Jan 2011
TL;DR: Weakconvergence methods in metric spaces were studied in this article, with applications sufficient to show their power and utility, and the results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables.
Abstract: The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

3,554 citations

01 Jan 2009
TL;DR: In this paper, a criterion for the convergence of numerical solutions of Navier-Stokes equations in two dimensions under steady conditions is given, which applies to all cases, of steady viscous flow in 2D.
Abstract: A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

1,568 citations

Book ChapterDOI
31 Oct 2006

1,424 citations