BMO and Morrey–Campanato estimates for stochastic convolutions and Schauder estimates for stochastic parabolic equations.

Abstract In this paper, we are aiming to prove several regularity results for the following stochastic fractional heat equations with additive noises d u t (x) = Δ α 2 u t (x) d t + g (t , x) d η t , u 0 = 0 , t ∈ (0 , T ] , x ∈ G , for a random field u : (t , x) ∈ [ 0 , T ] × G ↦ u (t , x) = : u t (x) ∈ R , where Δ α 2 : = − (− Δ) α 2 , α ∈ (0 , 2 ] , is the fractional Laplacian, T ∈ (0 , ∞) is arbitrarily fixed, G ⊂ R d is a bounded domain, g : [ 0 , T ] × G × Ω → R is a joint measurable coefficient, and η t , t ∈ [ 0 , ∞) , is either a Brownian motion or a Lévy process on a given filtered probability space (Ω , F , P ; { F t } t ∈ [ 0 , T ]). To this end, we derive the BMO estimates and Morrey–Campanato estimates, respectively, for stochastic singular integral operators arising from the equations concerned. Then, by utilizing the embedding theory between the Campanato space and the Hölder space, we establish the controllability of the norm of the space C θ , θ / 2 (D ¯) , where θ ≥ 0 , D ¯ = [ 0 , T ] × G ¯. With all these in hand, we are able to show that the q -th order BMO quasi-norm of the α q 0 -order derivative of the solution u is controlled by the norm of g under the condition that η t is a Lévy process. Finally, we derive the Schauder estimate for the p -moments of the solution of the above stochastic fractional heat equations driven by Lévy noise. [ABSTRACT FROM AUTHOR]