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$\rho \text{-}mixing$, β-mixing$\beta \text{-}mixing$ and α-mixing$\alpha \text{-}mixing$. Stationary diffusions that are ρ-mixing$\rho \text{-}mixing$ have mixing coefficients that decay exponentially to zero. When they fail to be ρ-mixing$\rho \text{-}mixing$, they are still β-mixing$\beta \text{-}mixing$ and α-mixing$\alpha \text{-}mixing$; 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.