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.