Stochastic Partial Differential Equations Driven by Fractional Brownian Motion: Well-Posedness and Numerical Methods

This dissertation establishes original well-posedness results and develops novel numerical methods for stochastic partial differential equations driven by fractional Brownian motion, contributing to the rapidly developing mathematical theory of rough path analysis and its intersection with numerical analysis of stochastic systems with long-range dependence. Stochastic partial differential equations driven by fractional Brownian motion are of growing importance in mathematical finance, hydrology, and environmental science due to the long-memory properties of fractional Brownian motion, which captures the persistent autocorrelation structures observed in financial time series, river discharge data, and atmospheric dynamics relevant to Nigerian rainfall modelling. The dissertation proves novel existence and uniqueness theorems for mild solutions of semi-linear stochastic heat and wave equations with fractional Brownian noise using the Malliavin calculus framework and pathwise stochastic integration theory developed by Gubinelli's rough path approach, extending existing results to the case of super-critical Hurst exponents below one-half. Pathwise regularity of solutions in Hölder function spaces is characterised, with optimal regularity exponents derived in terms of the Hurst parameter of the driving noise and the fractional Sobolev regularity of initial data. Galerkin spectral and finite element numerical approximation schemes are developed, and convergence rates in mean square and pathwise senses are rigorously established. Applications to stochastic modelling of rainfall-runoff dynamics in the Niger Basin are developed. Keywords: stochastic PDEs, fractional Brownian motion, rough path theory, Malliavin calculus, numerical methods