Nonlinearity and temporal dependence

Nonlinearities in the drift and diffusion coefficients influence temporal dependence in diffusion models. We study this link using three measures of temporal dependence: ρ-mixing, β-mixing and α-mixing. Stationary diffusions that are ρ-mixing have mixing coefficients that decay exponentially to zero. When they fail to be ρ-mixing, they are still β-mixing and α-mixing; but coefficient decay is slower than exponential. For such processes we find transformations of the Markov states that have finite variances but infinite spectral densities at frequency zero. The resulting spectral densities behave like those of stochastic processes with long memory. Finally we show how state dependent, Poisson sampling alters the temporal dependence.