Weak Approximations of Stochastic Partial Differential Equations with Fractional Noise (Journal of Computational Mathematics, 2024)
This paper aims to analyze the weak approximation error of a fully discrete scheme fora class of semi-linear parabolic stochastic partial differential equations (SPDEs) driven byadditive fractional Brownian motions with the Hurst parameter $H ∈ (1/2, 1).$ The spatialapproximation is performed by a spectral Galerkin method and the temporal discretizationby an exponential Euler method. As far as we know, the weak error analysis for approximations of fractional noise driven SPDEs is absent in the literature. A key difficulty inthe analysis is caused by the lack of the associated Kolmogorov equations. In the presentwork, a novel and efficient approach is presented to carry out the weak error analysis forthe approximations, which does not rely on the associated Kolmogorov equations but relieson the Malliavin calculus. To the best of our knowledge, the rates of weak convergence,shown to be higher than the strong convergence rates, are revealed in the fractional noisedriven SPDE setting for the first time. Numerical examples corroborate the claimed weakorders of convergence.
Keywords: Parabolic SPDEs; Fractional Brownian motion; Weak convergence rates; Spectral Galerkin method; Exponential Euler method; Malliavin calculus.