Existence and regularity of mild solutions to fractional stochastic evolution equations

This study is concerned with the stochastic fractional diffusion and diffusion-wave equations driven by multiplicative noise. We prove the existence and uniqueness of mild solutions to these equations by means of the Picard’s iteration method. With the help of the fractional calculus and stochastic analysis theory, we also establish the pathwise spatial-temporal (Sobolev-Hölder) regularity properties of mild solutions to these types of fractional SPDEs in a semigroup framework. Finally, we relate our results to the selection of appropriate numerical schemes for the solutions of these time-fractional SPDEs