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 ≥ 2 driven by a tempered fractional Brownian motion (TFBM) Bσ,λ(t) with −1/2 < σ < 0 and λ > 0. First, the global existence and unique ness of mild solutions are established by using a new technical lemma for stochastic integrals with respect to TFBM in the sense of p-th moment. Moreover, based on the relations between the stochastic integrals with respect to TFBM and fractional Browni an motion, we show the continuity of mild solutions in the case of λ → 0, σ ∈ (−1/2, 0) or λ > 0, σ → σ0 ∈ (−1/2, 0). In particular, we obtain p-th moment H¨older regularity in time and p-th 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.