Stochastic fractional diffusion equations containing finite and infinite delays with multiplicative noise

Abstract

In this work, we investigate stochastic fractional diffusion equations with Caputo–Fabrizio fractional derivatives and multiplicative noise, involving finite and infinite delays. Initially, the existence and uniqueness of mild solution in the spaces C p ( [ − a , b ] ; L q ( Ω , H ˙ r ) ) ) and C δ ( ( − ∞ , b ] ; L q ( Ω , H ˙ r ) ) ) are established. Next, besides investigating the regularity properties, we show the continuity of mild solutions with respect to the initial functions and the order of the fractional derivative for both cases of delay separately.

Continuity with respect to the Hurst parameter of solutions to stochastic evolution equations driven by H-valued fractional Brownian motion

“…In (H), we consider X 0 , f , g as functions taking value on the usual space L 2 (Ω, H) and L 2 (Ω, L 2 0 ) to guarantee the existence result on C([0, T ]; L 2 (Ω, H)), which makes our problem become simple and easy to be handled mathematically. The spaces can be extended to more complicated cases, for instance, L p (Ω, W k,l ) and L q (Ω, L 2 0 (H 1 , H 2 )) respectively (where W k,l is some Sobolev space, H 1 , H 2 are two Hilbert scale spaces) to ensure the existence of the solution on some Hölder continuity space C([0, T ]; L p ′ (Ω, W k ′ ,l ′ )) (see to [24] for an existence result for a SPED in this space). The strategy used to extend may be to apply some calculus inequalities, some Sobolev embeddings, and stochastic tools such as the Burkholder-Davis-Gundy-type inequality, the Kahane-Khintchine inequality, etc.…”