Stochastic partial differential equations and covariance matrices: analytical and geometric perspectives

This thesis investigates how geometric structure governs analytic and probabilistic phenomena in two complementary settings: stochastic partial differential equations (SPDEs) with multiple characteristics, and the geometry of low-rank covariance matrices. In the first part, we study linear hyperbolic SPDEs driven by Gaussian noise. We focus on three representative model operators, where the coefficients contain harmonic oscillators in the space variables, while the noise is additive, white in time and colored in space. We compute explicit fundamental solutions and then construct random-field solutions via stochastic convolution in the Walsh–Dalang framework. We derive sharp, geometric-dependent conditions on the spatial covariance of the noise (equivalently on the associated spectral measures) ensuring existence and measurability of the solution process. The analysis highlights how the symplectic geometry of the characteristic set and the presence of lower-order drift terms influence the admissible regularity and “color” of the driving noise. In the second part, we develop an associated-bundle description of the fixed-rank stratum of covariance matrices equipped with the Bures–Wasserstein metric. Working in this bundle picture, we (i) prove that the fibers are totally geodesic, (ii) derive a system of differential equations for Bures–Wasserstein geodesics, and (iii) establish a one-to-one correspondence between Grassmannian logarithms and Bures–Wasserstein logarithms on the fixed rank covariance matricees, and hence between minimizing geodesics in the two spaces. This alternative viewpoint clarifies the role of the underlying Grassmannian base and sets the stage for further investigations into structured covariance models.