Laura M. Morato

On the dynamics of diffusions and the related general electromagnetic potentials

In this paper the Nelson’s stochastic mechanics is extended to general diffusion motions. A representation theorem is proved which gives a one‐to‐one correspondence between solutions of certain Schrodinger equations and diffusion processes satisfying appropriate regularity conditions. Exploiting results of stochastic mechanics on Riemannian manifolds it is shown that the real part of the Schrodinger equations corresponding to the considered diffusions can be interpreted as Newton’s second law where the force is produced by generalized electromagnetic potentials.

Lagrangian variational principle in stochastic mechanics: Gauge structure and stability

The Lagrangian variational principle with the classical action leads, in stochastic mechanics, to Madelung’s fluid equations, if only irrotational velocity fields are allowed, while new dynamical equations arise if rotational velocity fields are also taken into account. The new equations are shown to be equivalent to the (gauge invariant) system of a Schrodinger equation involving a four‐vector potential (A,Φ) and the coupled evolution equation (of magnetohydrodynamical type) for the vector field A. A general energy theorem can be proved and the stability properties of irrotational and rotational solutions investigated.

Stochastic quantization for a system of N identical interacting Bose particles

We apply stochastic quantization to a system of N interacting identical bosons in an external potential Φ, by means of a general stationary-action principle. The collective motion is described in terms of a Markovian diffusion on , with joint density and entangled current velocity field , in principle of non-gradient form, related to one another by the continuity equation. Dynamical equations relax to those of canonical quantization, in some analogy with Parisi–Wu stochastic quantization. Thanks to the identity of particles, the one-particle marginal densities ρ, in the physical space , are all the same and it is possible to give, under mild conditions, a natural definition of the single-particle current velocity, which is related to ρ by the continuity equation in . The motion of single particles in the physical space comes to be described in terms of a non-Markovian three-dimensional diffusion with common density ρ and, at least at dynamical equilibrium, common current velocity v. The three-dimensional drift is perturbed by zero-mean terms depending on the whole configuration of the N-boson interacting system. Finally, we discuss in detail under which conditions the one-particle dynamical equations, which in their general form allow rotational perturbations, can be particularized, up to a change of variables, to the Gross–Pitaevskii equations.

Non-Symmetric Diffusions and Related Hamiltonians

We exhibit a unitary correspondence between (equivalence classes) of Hamiltonians H(Φ, A), involving electromagnetic potentials (A,Φ), and generators of ‘bi-directional’ Markov semigroups associated with non-symmetric diffusion processes.

Markov diffusions in comoving coordinates and stochastic quantization of the free relativistic spinless particle

We revisit the classical approach of comoving coordinates in relativistic hydrodynamics and we give a constructive proof for their global existence under suitable conditions, which is proper for stochastic quantization. We show that it is possible to assign stochastic kinematics for the free relativistic spinless particle as a Markov diffusion globally defined on M 4. Then introducing dynamics by means of a stochastic variational principle with Einstein’s action, we are lead to positive‐energy solutions of the Klein–Gordon equation. The procedure exhibits relativistic covariance properties.

A connection between quantum dynamics and approximation of Markov diffusions

It is observed that the existence of an attracting set in the class of solutions to the stochastic Lagrangian variational principle leads to a natural problem of convergence of diffusions in the Carlen class. It is then shown how dynamical properties enable one to prove some convergence results in the two‐dimensional Gaussian case.

On the identification by filtering techniques of a biological n-compartment model in which the transport rate parameters are assumed to be stochastic processes.

In this paper biological compartmental models are considered which take into account the intrinsic randomness of the transport rate parameters and their possible variability in time. An identification procedure is presented for the estimation of the stochastic processes representing the transport rate parameters of a compartmental model from a noisy input-output experiment. The problem is formulated in terms of nonlinear filtering. A simple model is discussed for the case in which the transport rate parameters are independent of each other. The possibility of testing possible important features of the behavior of the transport rate parameters is also evidenced.

Entangled states in stochastic mechanics

An axiomatization of the core part of stochastic mechanics (SM) is proposed and this scheme is discussed as a hidden variables theory. We work out in detail an example with entanglement and rigorously prove that SM and quantum mechanics agree in predicting all the observed correlations at different times.

Localization of Relative Entropy in Bose-Einstein Condensation of Trapped Interacting Bosons

We consider a system of interacting diffusions which is naturally associated to the ground state of the Hamiltonian of a system of N pairinteracting bosons and we give a detailed description of the phenomenon of the “localization of the relative entropy”. The method is based on peculiar rescaling properties of the mean energy functional

Dissipation caused by a vorticity field and generation of singularities in Madelung fluid

We consider a generalization of Madelung fluid equations, which was derived in the 1980s by means of a pathwise stochastic calculus of variations with the classical action functional. At variance with the original ones, the new equations allow us to consider velocity fields with vorticity. Such a vorticity causes dissipation of energy and it may concentrate, asymptotically, in the zeros of the density of the fluid. We study, by means of numerical methods, some Cauchy problems for the bidimensional symmetric harmonic oscillator and observe the generation of zeros of the density and concentration of the vorticity close to central lines and cylindrical sheets. Moreover, keeping the same initial data, we perturb the harmonic potential by a term proportional to the density of the fluid, thus obtaining an extension with vorticity of the Gross–Pitaevskii equation, and observe analogous behaviours.