Lack of regularity of the transport density in the monge problem

In this paper, we provide a family of counterexamples to the regularity of the transport density in the classical Monge-Kantorovich problem. We prove that the W^{1,p} regularity of the source and target measures f ^\pm does not imply that the transport density $\sigma$ is W^{1,p} , that the BV regularity of f ^\pm does not imply that $\sigma$ is BV and that f^\pm $\in$ C^\infty does not imply that $\sigma$ is W^{1,p} , for large p

On a class of stochastic differential equations used in quantum optics

Stochastic differential equations for processes with values in Hilbert spaces are now largely used in the quantum theory of open systems. In this work we present a class of such equations and discuss their main properties; moreover, we explain how they are derived from purely quantum mechanical models, where the dynamics is represented by a unitary evolution in a Hilbert space, and how they are related to the theory of continual measurements. An essential tool is an isomorphism between the bosonic Fock space and the Wiener space, which allow to connect certain quantum objects with probabilistic ones.Comment: 13 pages, LaTeX2

Stabilizing effect of large average initial velocity in forced dissipative PDEs invariant with respect to Galilean transformations

We describe a topological method to study the dynamics of dissipative PDEs on a torus with rapidly oscillating forcing terms. We show that a dissipative PDE, which is invariant with respect to Galilean transformations, with a large average initial velocity can be reduced to a problem with rapidly oscillating forcing terms. We apply the technique to the Burgers equation, and the incompressible 2D Navier-Stokes equations with a time-dependent forcing. We prove that for a large initial average speed the equation admits a bounded eternal solution, which attracts all other solutions forward in time. For the incompressible 3D Navier-Stokes equations we establish existence of a locally attracting solution

A Mechanism for Pockets of Predictability in Complex Adaptive Systems

We document a mechanism operating in complex adaptive systems leading to dynamical pockets of predictability (``prediction days''), in which agents collectively take predetermined courses of action, transiently decoupled from past history. We demonstrate and test it out-of-sample on synthetic minority and majority games as well as on real financial time series. The surprising large frequency of these prediction days implies a collective organization of agents and of their strategies which condense into transitional herding regimes.Comment: 5 pages, 3 figures, error correcte