arXiv subjects

Francesco Carlo De Vecchi

Publications and source records attributed to Francesco Carlo De Vecchi.

5 recordsLinked to original sources

arXiv2026

Two-Dimensional Euclidean Free Fermi Fields from Grassmannian Stochastic Analysis

Motivated by Nelson's Euclidean Markov-field construction, we develop a counterpart for the massive two-dimensional free Dirac field. Starting from a Gaussian white noise with values in a Grassmann algebra on a non-commutative probability space, we construct the associated Brownian motion and a stationary Ornstein--Uhlenbeck field. The resulting Euclidean Fermi field is realized as a tempered distribution on spacetime test functions, with values in the underlying algebra. Its mixed two-point functions reproduce the Euclidean Dirac fundamental solution. Time-zero fields are obtained by conditional expectation onto the past filtration generated by the noise, while their covariance is the reflected Euclidean Dirac kernel. We show that conditional expectation is compatible with Wick ordering and maps Euclidean Wick polynomials to the corresponding time-zero Wick polynomials. The Euclidean time-translation semigroup is then constructed on time-zero fields and Wick sectors, with generators given by the one-particle Hamiltonian and by its second quantization.

math-ph↗

arXiv2026

On the relation between transformations of SDEs and of associated Kolmogorov equations

The relationship between transformations of one-dimensional stochastic differential equations and those of the associated Kolmogorov backward equations is investigated. Stochastic transformations combining deterministic time changes, space-time diffeomorphisms, and Girsanov-type changes of probability measure are shown to be in correspondence with admissible transformations of the associated linear parabolic equations, namely the transformations preserving the parabolic structure up to the possible introduction of a zero-order potential term. A distinguished subclass of transformations, characterized by the absence of any potential term in the transformed Kolmogorov equation, is shown to coincide exactly with Doob transformations of the underlying probability measure of the SDE. This characterization further establishes a correspondence between Doob symmetries of an SDE and Lie point symmetries of the associated Kolmogorov backward equation. This correspondence, in turn, yields a constructive (probabilistic) Lie symmetry classification of autonomous Kolmogorov backward equations.

math.PR↗

arXiv2026

A criterion for proving entropy chaos on path space

A criterion for proving a strong form of propagation of chaos on the path space, known as entropy chaos, for a general interacting diffusion system is proposed. Our analysis focuses on the class of conservative diffusions introduced by Carlen, which are characterized by infinitesimal characteristic pairs, that is, a time-marginal probability density and a current velocity field. A key property of this broad class is that the processes remain diffusions under time-reversal. We prove that, given a suitable bound on the relative entropy (with respect to the Wiener measure) and the weak convergence of both drifts and fixed-time marginal densities, strong entropy chaos at the process level is achieved in the infinite particle limit, provided the limit drift satisfies a specific regularity condition. This stochastic framework encompasses various singular interacting particle systems and their related asymptotic scenarios.

math.PR↗

arXiv2025

Discrete reaction-diffusion system with stochastic dynamical boundary conditions: convergence results

A space discrete approximation to a highly nonlinear reaction-diffusion system endowed with a stochastic dynamical boundary condition is analyzed and the convergence of the discrete scheme to the solution to the corresponding continuum random system is established. A splitting strategy allows us to decompose the random system into a space-discrete heat equation with a stochastic boundary condition, and a nonlinear and nonlocal space-discrete differential system coupled with the first one and with deterministic initial and boundary conditions. The convergence result is obtained by first establishing some a priori estimates for both space-discrete splitted variables and then exploiting compact embedding theorems for time-space Besov spaces on the positive lattice. The convergence of a fully discrete approximation of the random system is also discussed.

math.PR↗

arXiv2025

Stochastic Hartree NLS in 3d coming from a Many-Body Quantum System with White Noise Potential

In this paper, we consider the defocusing Hartree NLS with white noise external potential on T^3 i.e. the Hartree NLS whose linear part is given by the Anderson Hamiltonian. A Strichartz-type estimate is established for the Anderson Hamiltonian using perturbative arguments and the local and global well-posedness of the NLS is considered with initial data in the domain and form-domain of the Anderson Hamiltonian under different regularity assumptions on the Hartree interaction. Furthermore, we establish the Anderson Hartree NLS as an effective equation describing many-body Bosonic systems and, in particular, we prove the convergence of the linear Schrödinger equation for the many body system to the BBGKY hierarchy for the Coulomb interaction.

math.AP↗