I am interested in a biological system that can be represented by an unknown stochastic dynamical system, and that generates a signal which can be measured. The measured signal is found to only contain linear correlations (this can be shown for instance by comparison with linear surrogates, see e.g. a review by Schreiber and Schmitz). What can be said about the linearity/non-linearity of the deterministic vector field of the underlying stochastic dynamical system?
The simplest possibility is that the deterministic vector field of the dynamical system is linear, or at least that the signal stays in a part of phase space where the deterministic vector field is locally linear. However it seems that for a deterministic system, the absence of non-linear correlations in the generated signal does not necessarily imply even local linearity of the vector field. For example a non-linear Stuart-Landau oscillator on its limit cycle generates sinusoids. What can be said for a stochastic dynamical system? Schreiber and Schmitz write "it could also be [...] that the process is nonlinear but the single time series at this sampling covers such a poor fraction of the rich dynamics that it must appear linear stochastic to the analysis." Can the statement be made more precise? Is it the case that assuming "sufficient" dynamical noise, the absence of non-linear structure in the measured signal implies that the deterministic vector field is at least locally linear?