Abstract. In this paper we introduce and analyze a stochastic particle method for the McKean-Vlasov and the Burgers equation; the construction and error analysis are based upon the theory of the propagation of chaos for interacting particle systems.Our objective is three-fold. First, we consider a McKean-Vlasov equation in [0, T ] × R with sufficiently smooth kernels, and the PDEs giving the distribution function and the density of the measure µt, the solution to the McKean-Vlasov equation. The simulation of the stochastic system with N particles provides a discrete measure which approximates µ k∆t for each time k∆t (where ∆t is a discretization step of the time interval [0, T ]). An integration (resp. smoothing) of this discrete measure provides approximations of the distribution function (resp. density) of µ k∆t . We show that the convergencetive distribution function at time T , and of order O ε 2 + 1 εof the density at time T (Ω is the underlying probability space, ε is a smoothing parameter). Our second objective is to show that our particle method can be modified to solve the Burgers equation with a nonmonotonic initial condition, without modifying the convergence rateThis part extends earlier work of ours, where we have limited ourselves to monotonic initial conditions. Finally, we present numerical experiments which confirm our theoretical estimates and illustrate the numerical efficiency of the method when the viscosity coefficient is very small.

Abstract. We consider one-dimensional stochastic differential equations in the particular case of diffusion coefficient functions of the form |x| α , α ∈ [1/2, 1). In that case, we study the rate of convergence of a symmetrized version of the Euler scheme. This symmetrized version is easy to simulate on a computer. We prove its strong convergence and obtain the same rate of convergence as when the coefficients are Lipschitz.Mathematics Subject Classification. 65C30, 60H35, 65C20.

In this paper we construct a stochastic particle method for the Burgers equation with a monotone initial condition; we prove that the

In this note, we clarify the well-posedness of the limit equations to the mean-field N-neuron models proposed in (Baladron et al. in J. Math. Neurosci. 2:10, 2012) and we prove the associated propagation of chaos property. We also complete the modeling issue in (Baladron et al. in J. Math. Neurosci. 2:10, 2012) by discussing the well-posedness of the stochastic differential equations which govern the behavior of the ion channels and the amount of available neurotransmitters.

In this article, we analyse the error induced by the Euler scheme combined with a symmetry procedure near the boundary for the simulation of diffusion processes with an oblique reflection on a smooth boundary. This procedure is easy to implement and, in addition, accurate: indeed, we prove that it yields a weak rate of convergence of order 1 with respect to the time-discretization step.

Abstract.Motivated by the development of efficient Monte Carlo methods for PDE models in molecular dynamics, we establish a new probabilistic interpretation of a family of divergence form operators with discontinuous coefficients at the interface of two open subsets of R d . This family of operators includes the case of the linearized Poisson-Boltzmann equation used to compute the electrostatic free energy of a molecule. More precisely, we explicitly construct a Markov process whose infinitesimal generator belongs to this family, as the solution of a SDE including a non standard local time term related to the interface of discontinuity. We then prove an extended Feynman-Kac formula for the Poisson-Boltzmann equation. This formula allows us to justify various probabilistic numerical methods to approximate the free energy of a molecule. We analyse the convergence rate of these simulation procedures and numerically compare them on idealized molecules models.Mathematics Subject Classification. 35Q60, 92C40, 60J60, 65C05, 65C20, 68U20.

This paper is motivated by a new class of SDEs-PDEs systems, the so called Lagrangian stochastic models which are commonly used in the simulation of turbulent flows. We study a position-velocity system which is nonlinear in the sense of McKean. As the dynamics of the velocity depends on the conditional expectation w.r.t. its position, the interaction kernel is singular.We prove existence and uniqueness of the solution to the system by solving a nonlinear martingale problem and showing that the corresponding interacting particle system propagates chaos.

In this article, we propose a new stochastic downscaling method: provided a numerical prediction of wind at large scale, we aim to improve the approximation at small scales thanks to a local stochastic model. We first recall the framework of a Lagrangian stochastic model borrowed from S.B. Pope. Then, we adapt it to our meteorological framework, both from the theoretical and numerical viewpoints. Finally, we present some promising numerical results corresponding to the simulation of wind over the Mediterranean Sea.