Spatial Discretization for Stochastic Semilinear Superdiffusion Driven by Fractionally Integrated Multiplicative Space–Time White Noise

Abstract

We investigate the spatial discretization of a stochastic semilinear superdiffusion problem driven by fractionally integrated multiplicative space–time white noise. The white noise is characterized by its properties of being white in both space and time, and the time fractional derivative is considered in the Caputo sense with an order α∈ (1, 2). A spatial discretization scheme is introduced by approximating the space–time white noise with the Euler method in the spatial direction and approximating the second-order space derivative with the central difference scheme. By using the Green functions, we obtain both exact and approximate solutions for the proposed problem. The regularities of both the exact and approximate solutions are studied, and the optimal error estimates that depend on the smoothness of the initial values are established.

Keywords

• superdiffusion
• integrated multiplicative noise
• Caputo derivative
• finite difference method
• optimal error estimates