We propose a new scheme for the long time approximation of a diffusion when the drift vector field is not globally Lipschitz. Under this assumption, regular explicit Euler scheme –with constant or decreasing step– may explode and implicit Euler scheme are CPU-time expensive. The algorithm we introduce is explicit and we prove that any weak limit of the weighted empirical measures of this scheme...

We analyze the Euler approximation to a state constrained control problem. We show that if the active constraints satisfy an independence condition and the Lagrangian satisfies a coercivity condition, then locally there exists a solution to the Euler discretization, and the error is bounded by a constant times the mesh size. The proof couples recent stability results for state constrained contr...

Equations of motion of an axially symmetric sphere rolling and sliding on a plane are usually taken as model of the tippe top. We study these equations in the nonsliding regime both in the vector notation and in the Euler angle variables when they admit three integrals of motion that are linear and quadratic in momenta. In the Euler angle variables (θ, φ, ψ) these integrals give separation equa...

We define a homomorphism ζ from the algebra of quasi-symmetric functions to the reals which involves the Euler constant and multiple zeta values. Besides advancing the study of multiple zeta values, the homomorphism ζ appears in connection with two Hirzebruch genera of almost complex manifolds: the Γ-genus (related to mirror symmetry) and the Γ̂-genus (related to an S1-equivariant Euler class). ...

The main goal of the asymptotic equivalence theory of Le Cam (1986) is to approximate general statistical models by simple ones. We develop here a global asymptotic equivalence result for nonparametric drift estimation of a discretely observed diffusion process and its Euler scheme. The asymptotic equivalences are established by constructing explicit equivalence mappings. The impact of such asy...

We study the local-in-time hydrodynamic limit of the relativistic Boltzmann equation using a Hilbert expansion. More specifically, we prove the existence of local solutions to the relativistic Boltzmann equation that are nearby the local relativistic Maxwellian constructed from a class of solutions to the relativistic Euler equations that includes a large subclass of near-constant, non-vacuum f...

We study the weak approximation error of a skew diffusion with bounded measurable drift and Hölder diffusion coefficient by an Euler-type scheme, which consists of iteratively simulating skew Brownian motions with constant drift. We first establish two sided Gaussian bounds for the density of this approximation scheme. Then, a bound for the difference between the densities of the skew diffusion...

Consider the p-system describing the subsonic flow of a fluid in a pipe with section a = a(x). We analyze the mathematical problem related to a junction, i.e., a sharp discontinuity in the pipe’s geometry, we consider the case of a picewise constant pipe’s section and then, the smooth case. In particular, through a limit procedure, we prove the well posedness of the smooth case from the discont...

Let G be a graph of n vertices with maximum degree d that can be drawn without crossing in a closed surface of Euler characteristic χ. It is proved that then G can be drawn in the plane with at most cχdn crossings, where cχ is a constant depending only on χ. This result, which is tight up to a constant factor, is strengthened and generalized to the case when there is no restriction on the degre...

We consider a continuous family (fs), s ∈ [0, 1] of complex polynomials in two variables with isolated singularities, that are Newton non-degenerate. We suppose that the Euler characteristic of a generic fiber is constant. We firstly prove that the set of critical values at infinity depends continuously on s, and secondly that the degree of the fs is constant (up to an algebraic automorphism of...