On a stochastic partial differential equation with a fractional Laplacian operator

In this article, we consider the regularity of the solution of d u ( t , x ) = ( Δ α 2 u ( t , x ) + f ( t , x ) ) d t + ∑ i = 1 m g i ( t , x ) d w t i , u ( 0 , x ) = u 0 ( x ) . We adopt the framework given in some works of Krylov which are related to the theory of stochastic partial differential equations with the Laplace operator. We construct the important estimates for the theory and prove regularity theorems using them.