arXiv subjects

Leonardo Marconi

Publications and source records attributed to Leonardo Marconi.

3 recordsLinked to original sources

arXiv2026

Geometric Regime--Switching Diffusions on Stratified Riemannian Spaces with an Application to Covariance Matrices

We construct geometric regime-switching diffusions, a class of Markov processes on locally compact stratified Riemannian state spaces. In contrast with classical regime-switching and stochastic hybrid diffusions, the regimes are not external labels, but intrinsic strata of a singular geometric state space. Changes of regime may therefore change dimension, rank, geometry or combinatorial type while the state space maintains its ambient topology. On each stratum the motion is a conservative Feller diffusion, while inter-stratum transitions are specified by state-dependent jump rates and landing kernels along a directed graph. We characterize the process through a martingale problem on a natural stratified core. Under a uniform bound on the total jump rate, we construct a conservative c\`adl\`ag strong Markov process by combining the stratumwise diffusions with a Poisson thinning mechanism. Uniqueness is proved using an auxiliary disjoint-union topology and a bounded perturbation argument. Standard Foster--Lyapunov conditions for the extended generator give positive Harris recurrence, uniqueness of the invariant probability measure and, under aperiodicity, $V$-uniform geometric ergodicity. The framework is applied to the cone of positive semidefinite covariance matrices, stratified by rank. The resulting process combines fixed-rank covariance diffusions with stochastic rank changes and is $V$-uniformly geometrically ergodic.

math.PR↗

arXiv2026

An associated bundle approach to the Bures--Wasserstein geometry of fixed rank covariance matrices

The Bures--Wasserstein geometry of covariance matrices provides a canonical distance on the statistical manifold of centred Gaussian measures and lies at the intersection of information geometry, quantum information, and optimal transport. The space of covariance matrices admits a natural stratified structure whose strata consist of fixed-rank covariance matrices. In this paper we focus on the rank-$k$ stratum $\Sym^+(n,k)$ and revisit its geometry through the diffeomorphic associated-bundle model $\Sym^+(n,k)\cong\St(n,k)\times_{O(k)}\Sym^{+}(k)$. Working in this bundle picture, we (i) derive a system of differential equations for Bures--Wasserstein geodesics, (ii) prove that the fibers are totally geodesic and (iii) establish a one-to-one correspondence between Grassmannian logarithms and Bures--Wasserstein logarithms on $\Sym^+(n,k)$, and hence between minimizing geodesics in the two spaces. This alternative viewpoint clarifies the role of the underlying base $\Gr(k,n)$ in the Bures--Wasserstein geometry of low-rank covariance matrices and sets the stage for further investigations into structured covariance models.

math.DG↗

arXiv2024

On some stochastic hyperbolic equations with symplectic characteristics

We study the effect of Gaussian perturbations on a class of model hyperbolic partial differential equations with double symplectic characteristics in low spatial dimensions, extending some recent work in [5]. The coefficients of our partial differential operators contain harmonic oscillators in the space variables, while the noise is additive, white in time and colored in space. We provide sufficient conditions on the spectral measure of the covariance functional describing the noise that allows for the existence of a random field solution for the resulting stochastic partial differential equation. Furthermore we show how the symplectic structure of the set of multiple points affects the regularity of the noise needed to build a measurable process solution. Our approach is based on some explicit computations for the fundamental solutions of several model partial differential operators together with their explicit Fourier transforms.

math.PR↗