Pullback and forward attractors for dissipative cellular neural networks with additive noises

This work investigates the dissipative dynamical system in the infinite lattice Z with cellular neural networks as an example of application. The dynamics of each node depends on itself and nearby nodes by a nonlinear function. When each node is perturbed with weighted Gaussian white noise, there exists a unique pullback attractor and forward attractor whose domain of attraction are random tempered sets. Furthermore, we prove that the pullback and forward attractor are equivalent to a random equilibrium which is also tempered. Both convergence to the pullback and forward attractors are exponentially fast.