Q-fractional Brownian motion and Lévy-driven SPDEs on the sphere: analysis and simulation

Licentiate thesis, 2026

Many real-world phenomena can be modeled by stochastic processes in time, space or space-time. In important application fields like climate modeling and cosmology, the underlying spatial domain is a sphere, which represents the Earth. This thesis is concerned with spatio-temporal stochastic processes and stochastic partial differential equations on the unit sphere and is based on two papers. In the first paper, isotropic Q-fractional Brownian motion is discussed, which models isotropic random fields that evolve over time according to a fractional Brownian motion. This Q-fractional Brownian motion has applications, for example, in modeling the cosmic microwave background. The spatio-temporal Hölder regularity of Q-fractional Brownian motion is analyzed and efficient numerical methods for its simulation are investigated. The second paper focuses on the long-time behavior of stochastic partial differential equations driven by Q-Lévy noise on the sphere. For the linear stochastic wave, Schrödinger and Maxwell's equations, physical quantities such as, for example, energy, that are conserved in the deterministic case, are considered. It is proved that under additive noise, the expectations of these quantities grow linearly following trace formulas. Numerical discretization schemes are analyzed and it is proved that exponential Euler-type schemes reproduce the trace formulas while forward and backward Euler-Maruyama methods fail to do so. Extensive numerical experiments illustrate these results.