Stefania Ugolini

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.

Symmetries of stochastic differential equations: A geometric approach

A new notion of stochastic transformation is proposed and applied to the study of both weak and strong symmetries of stochastic differential equations (SDEs). The correspondence between an algebra of weak symmetries for a given SDE and an algebra of strong symmetries for a modified SDE is proved under suitable regularity assumptions. This general approach is applied to a stochastic version of a two dimensional symmetric ordinary differential equation and to the case of two dimensional Brownian motion.

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.

Reduction and reconstruction of stochastic differential equations via symmetries

An algorithmic method to exploit a general class of infinitesimal symmetries for reducing stochastic differential equations is presented, and a natural definition of reconstruction, inspired by the classical reconstruction by quadratures, is proposed. As a side result, the well-known solution formula for linear one-dimensional stochastic differential equations is obtained within this symmetry approach. The complete procedure is applied to several examples with both theoretical and applied relevance.

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

Convergence of Nelson diffusions with time‐dependent electromagnetic potentials

Some recent results on the convergence of Nelson diffusions are extended to the case of Schrodinger operators with time‐dependent electromagnetic potentials. It is proven that the sequence {P n}n≥1 of measures on the canonical space of physical trajectories associated to the solutions of Schrodinger equations in Nelson’s scheme, corresponding to the sequence {(Vn,An)}n≥1 ⊆C1(R;R×L2(R3)), converges in the total variation norm under the assumptions that for every fixed t the scalar potentials Vn(t) converge in R, the space of Rollnik class potentials, and the vector potentials An(t) converge in Lloc∞(R;L2(R3)). In order to prove these results conditions are given under which solutions of Schrodinger equations are continuous in the (time‐dependent electromagnetic) potentials in the norm of the Sobolev space H1(R3).

Scattering into cones and flux across surfaces in quantum mechanics: A pathwise probabilistic approach

We show how the scattering-into-cones and flux-across-surfaces theorems in quantum mechanics have very intuitive pathwise probabilistic versions based on some results by Carlen about large time behavior of paths of Nelson’s diffusions. The quantum mechanical results can then be recovered by taking expectations in our pathwise statements.

Complex Dirichlet Forms: Non Symmetric Diffusion Processes and Schrödinger Operators

The concept of complex Dirichlet forms εc resp. operators Lc in complex weighted L2-spaces is introduced. Perturbations of classical Dirichlet forms by forms associated with complex first-order differential operators provide examples of complex Dirichlet forms.Complex Dirichlet operators Lc are unitarily equivalent with (a family of) Schrödinger operators with electromagnetic potentials.To εc there is associated a pair of real-valued non symmetric Dirichlet forms on the corresponding real weighted L2-spaces, which in turn are associated with (non-symmetric) diffusion processes.Results by Stannat on non symmetric Dirichlet forms and their perturbations can be used for discussing the essential self-adjointness of Lc.New closability criteria for (perturbation of) non symmetric Dirichlet forms are obtained.

Entropy Chaos and Bose-Einstein Condensation

We prove the entropy-chaos property for the system of N indistinguishable interacting diffusions rigorously associated with the ground state of N trapped Bose particles in the Gross–Pitaevskii scaling limit of infinitely many particles. On the path-space we show that the sequence of probability measures of the one-particle interacting diffusion weakly converges to a limit probability measure, uniquely associated with the minimizer of the Gross-Pitaevskii functional.

Stochastic quantization of finite dimensional systems with electromagnetic interactions

We study the stochastic quantization of finite dimensional systems via path-wise calculus of variations with the mean discretized classical action in the general case of electromagnetic interactions. We show that there exists a unique choice of the mean discretized action corresponding to the minimal classical magnetic coupling and derive the general equations of motion by means of a path-wise stochastic calculus of variations. In the case of purely scalar interactions, the total mean energy of the system (which gives the usual quantum mechanical expectation of the Hamiltonian in the canonical limit) works as a Lyapunov functional and the system relaxes on the canonical solutions, represented by Nelsons diffusions, which act as an attracting set. We show that, in presence of a minimal magnetic coupling, the mean energy is no longer a Lyapunov functional. We construct for a simple example, a new Lyapunov functional, and we show that the system can reach the dynamical equilibrium also by absorbing energy from the external magnetic field.