Delayed stochastic systems

Noise and time delay are two elements that are associated with many natural systems, and often they are sources of complex behaviors. Understanding of this complexity is yet to be explored, particularly when both elements are present. As a step to gain insight into such complexity for a system with both noise and delay, we investigate such delayed stochastic systems both i n dynamical and probabilistic perspectives. A Langevin equation with delay and a random-walk model whose transition probability depends on a fixed tim e-interval past (delayed random walk model) are the subjects of in depth fo cus. As well as considering relations between these two types of models, we derive an approximate Fokker-Planck equation for delayed stochastic system s and compare its solution with numerical results.