A regularity theory for stochastic partial differential equations driven by multiplicative space-time white noise with the random fractional Laplacians

We present the existence, uniqueness, and regularity of a strong solution to a super-linear stochastic partial differential equation (SPDE) with the random fractional Laplacians: u t = a Δ α / 2 u + b u x + c u + ξ | u | 1 + λ W ˙ , t > 0 ; u ( 0 , · ) = u 0 ( · ) , where W ˙ is a space-time white noise, α ∈ ( 1 , 2 ) , and λ ∈ [ 0 , α / 2 - 1 / 2 ) . The leading coefficient a satisfies the ellipticity condition and depends on ( ω , t ) . The lower-order coefficients b , c , and ξ depend on ( ω , t , x ) . The coefficients a , b , c , and ξ are bounded. The initial data u 0 depends on ( ω , x ) . The unique existence of local solutions to the SPDE follows from the unique solvability of a general Lipschitz case. We prove a Hölder embedding theorem for solution space H p γ ( τ ) and maximum principle for SPDEs with the random fractional Laplacians to extend local solutions to a global one. The range of λ ∈ [ 0 , α / 2 - 1 / 2 ) depending on the highest order of the fractional Laplacian is given as a sufficient condition for the existence. When α ↑ 2 , the condition is in accordance with the one for unique solvability of Laplacian case. Moreover, the Hölder embedding theorem provides maximal Hölder regularity of the solution u ( ω , t , x ) , which has α times as much regularity in space as in time; for T ∈ ( 0 , ∞ ) and small ε > 0 , almost surely u ∈ C t , x 1 2 - 1 2 α - λ α - ε , α 2 - 1 2 - λ - ε ( [ 0 , T ] × R ) .