Well-posedness and Mittag–Leffler Euler integrator for space-time fractional SPDEs with fractionally integrated additive noise
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This paper considers the space-time fractional stochastic partial differential equation (SPDE, for short) with fractionally integrated additive noise, which is general and includes many (fractional) SPDEs with additive noise. Firstly, the existence, uniqueness and temporal regularity of the mild solution are presented. Then the Mittag–Leffler Euler integrator is proposed as a time-stepping method to numerically solve the underlying model. Two key ingredients are developed to overcome the difficulty caused by the interaction between the time-fractional derivative and the fractionally integrated noise. One is a novel decomposition way for the drift part of the mild solution, named here the integral decomposition technique, and the other is to derive some fine estimates associated with the solution operator by making use of the properties of the Mittag–Leffler function. Consequently, the proposed Mittag–Leffler Euler integrator is proved to be convergent with order min{1/2 + α/2β(r+λ-κ), 1 } if α + γ = 1, otherwise order min{α/2βmin{κ,r+λ} + (γ-1/2)^+, 1-ε} in the sense of L^2(Ω,H)-norm for the nonlinear case. In particular, the corresponding convergence order can attain min{α + α r/2β + (γ-1/2)^+-ε, α+γ -ε, 1 } for the linear case.
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