A. Hernández-Machado

Joint probability distribution of nonmarkovian SDE

We obtain a time convolutionless partial differential equation for the two events joint probability distribution of nonmarkovian processes defined by stochastic differential equations with colored noise. As an example we discuss nonmarkovian brownian motion.

Dynamical properties of non‐Markovian stochastic differential equations

We study nonstationary non‐Markovian processes defined by Langevin‐type stochastic differential equations with an Ornstein–Uhlenbeck driving force. We concentrate on the long time limit of the dynamical evolution. We derive an approximate equation for the correlation function of a nonlinear nonstationary non‐Markovian process, and we discuss its consequences. Non‐Markovicity can introduce a dependence on noise parameters in the dynamics of the correlation function in cases in which it becomes independent of these parameters in the Markovian limit. Several examples are discussed in which the relaxation time increases with respect to the Markovian limit. For a Brownian harmonic oscillator with fluctuating frequency, the non‐Markovicity of the process decreases the domain of stability of the system, and it can change an infradamped evolution into an overdamped one.

Correlation functions of non-linear nuclear reactor models

Abstract The steady-state correlation function of a non-linear point nuclear reactor model with instantaneous temperature feedback, reactivity fluctuations and neglection of delayed neutrons is calculated by means of a continued fraction expansion method. Near criticality, where other methods fail, this method gives good results for early times. A first-order truncation gives a spectral density with two corner frequencies. Results are given for a non-vanishing independent neutron source. In this case no exact results are known. For a vanishing source our results are compared with exact ones and with those of a linear analysis and a system size expansion. The necessity of a proper treatment of non-Markovian effects for non-white reactivity fluctuations is pointed out. The existence, in the spectral density, of a corner frequency associated with the correlation time of the noise is shown.

Correlation functions near instabilities in systems driven by parametric noise

The steady-state correlation functions of non-linear stochastic processes driven by parametric noise are studied. A systematic method proposed by Nadler and Schulten is applied beyond the lowest order for the first time in this context. It is reformulated in a way which admits generalization to noises other than Gaussian and white. The explicit results obtained close to the instability point for the Verhulst model with Gaussian white noise improve considerably those previously obtained by continued-fraction techniques.