Let B = (Bt)t≥0 be a standard Brownian motion started at zero, and let μ ∈ R be a given and fixed constant. Set B t = Bt+μt and S t = max 0≤s≤t B s for t ≥ 0 . Then the process: (x ∨ Sμ)−Bμ = ((x ∨ S t )−B t )t≥0 realises an explicit construction of the reflecting Brownian motion with drift −μ started at x in R+ . Moreover, if the latter process is denoted by Z = (Z t )t≥0 , then the classic Lé...

We investigate the spectral asymptotic properties of the stationary dynamical system ξt = φ(T (X0)). This process is given by the iterations of a piecewise expanding map T of the interval [0, 1], invariant for an ergodic probability μ. The initial state X0 is distributed over [0, 1] according to μ and φ is a function taking values in R. We establish a strong law of large numbers and a central l...

A semiclassical transport equation for electrons in a semiconductor crystal is considered. It includes a nonlinear scattering operator modelling electron-phonon interaction. A formal macroscopic limit under the assumption of a small phonon energy is carried out, leading to the so called spherical harmonics expansion (SHE-) model for an energy-dependent distribution function with a scattering op...

In this paper, we study optimal control problems associated with a scalar hyperbolic conservation law modeling the development of ovarian follicles. Changes in the age and maturity of follicular cells are described by a 2D conservation law, where the control terms act on the velocities. The control problem consists in optimizing the follicular cell resources so that the follicular maturity reac...

We introduce a new nonconservative self-organized critical model. This model is equivalent to a quasistatic two-dimensional version of the Burridge-KnopoA spring-block model of earthquakes. Our model displays a robust power-law behavior. The exponent is not universal; rather it depends on the level of conservation. A dynamical phase transition from localized to nonlocalized behavior is seen as ...

We explore the connections between the constraints on the precision of quantum logical operations that arise from a conservation law, and those arising from quantum field fluctuations. We show that the conservation-law based constraints apply in a number of situations of experimental interest, such as Raman excitations, and atoms in free space interacting with the multimode vacuum. We also show...

This paper contains a qualitative study of a scalar conservation law with viscosity: ut + f(u)x = uxx . We consider the problem of identifying the location of viscous shocks, thus obtaining an optimal finite dimensional description of solutions to the viscous conservation law. We introduce a nonlinear functional whose minimizers yield the viscous travelling profiles which “optimally fit” the gi...

We give a stochastic algorithm to compute the p-mean of a probability measure μ whose support is contained in a regular geodesic ball of a Riemannian manifold. We show the almost sure convergence of this algorithm. Under a further regularity condition, an associated central limite theorem is also proved. Moreover, it is shown that a deterministic subgradient algorithm can also be used to comput...

We consider a strictly hyperbolic, genuinely nonlinear system of conservation laws in one space dimension. A sharp decay estimate is proved for the positive waves in an entropy weak solution. The result is stated in terms of a partial ordering among positive measures, using symmetric rearrangements and a comparison with a solution of Burgers’ equation with impulsive sources. 1 Introduction Cons...