The continuity, regularity and polynomial stability of mild solutions for stochastic 2D-Stokes equations with unbounded delay driven by tempered fractional Gaussian noise
Abstract
We consider stochastic 2D-Stokes equations with unbounded delay in fractional power spaces and moments of order p≥2p≥2 driven by a tempered fractional Brownian motion (TFBM) Bσ,λ(t)Bσ,λ(t) with −1/2<σ<0−1/2<σ<0 and λ>0λ>0. First, the global existence and uniqueness of mild solutions are established by using a new technical lemma for stochastic integrals with respect to TFBM in the sense of ppth moment. Moreover, based on the relations between the stochastic integrals with respect to TFBM and fractional Brownian motion, we show the continuity of mild solutions in the case of λ→0λ→0, σ∈(−1/2,0)σ∈(−1/2,0) or λ>0λ>0, σ→σ0∈(−1/2,0)σ→σ0∈(−1/2,0). In particular, we obtain ppth moment Hölder regularity in time and ppth polynomial stability of mild solutions. This paper can be regarded as a first step to study the challenging model: stochastic 2D-Navier–Stokes equations with unbounded delay driven by tempered fractional Gaussian noise.
