Strong error analysis of a stochastic exponential integrator for SPDEs driven by fractional Brownian motion with H<1/2

Source abstract

Fractional Brownian motion provides a useful framework for modeling random phenomena with anti-persistent temporal correlations that arise in applications such as anomalous diffusion, hydrology, finance, and geophysical processes. In this paper, we study a class of semilinear stochastic partial differential equations driven by additive fractional Brownian motion with Hurst parameter $H\in(0,\frac12)$. We establish the well-posedness and space-time regularity of the mild solution and investigate the strong convergence of a stochastic exponential integrator for the temporal discretization. The analysis allows the linear operator to be non-self-adjoint and combines analytic semigroup estimates with the canonical Hilbert-space structure associated with fractional Brownian motion and Malliavin calculus to handle the low temporal regularity of the noise. We derive an $H$-dependent strong temporal convergence rate, which reaches $H+\frac12$ under maximal spatial regularity. Numerical experiments for a stochastic advection--diffusion--reaction problem with heterogeneous Darcy flow confirm the theoretical temporal convergence rates. We also estimate the mean of the solution with different Hurst parameters $H\in(0,\frac12)$ using the Monte Carlo method.