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 [Formula: see text] driven by a tempered fractional Brownian motion (TFBM) [Formula: see text] with [Formula: see text] and [Formula: see text]. 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 [Formula: see text]th 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 [Formula: see text], [Formula: see text] or [Formula: see text], [Formula: see text]. In particular, we obtain [Formula: see text]th moment Hölder regularity in time and [Formula: see text]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.