Nonlocal stochastic functional differential equations driven by G‐Brownian motion and mean random dynamical systems

Abstract

In this paper, we consider a class of nonlocal stochastic functional differential equations driven by G‐Brownian motion (GNSFDEs) at phase space whose coefficients are dependent on the p‐th moment. Existence and uniqueness of solutions for GNSFDEs are investigated by virtue of theory of nonlinear expectation. Furthermore, we show that the solution map of GNSFDEs can generate a p‐mean random dynamical system in nonlinear expectation framework. In addition, two examples are provided to illustrate the effectiveness of the obtained results

$G$-mean random attractors for complex Ginzburg–Landau equations with probability-uncertain initial data

“…. In order to investigate the asymptotic behavior of system (1.1) with random initial conditions, following the idea of [9,19,34], we will introduce the concepts of G-mean random dynamical system and pullback G-mean random attractors over (Ω,H, Ê,F) (not over (Ω,H, Ê,F,F t )). For convenience, let…”

“…Stability has also been extensively studied in the literature, see, e.g., [10,22,26]. In addition, in [9], the mean random dynamical system was introduced in nonlinear expectation framework, which can be used for nonlocal stochastic systems driven by G-Brownian motion.…”

“…(iii) Different from [9], the concept of G-mean random dynamical system is introduced in L 2 G (Ω,L 2 (R)) over (Ω,H, Ê,F), instead of over a filtered space (Ω,H, Ê,F,F t ). Meanwhile, the definition of pullback mean random attractor for G-mean random dynamical system is given.…”