Abstract
Stochastic differential equations describe a large variety of physical phenomena. The noise appearing in this equations models the influence of external fields or the interaction with internal subsystems. Quantities of interest are the stationary probability density, moments and correlation functions, and asymptotic properties, e.g. the decay of correlations. Qualitative changes in this quantities caused by the noise are called noise induced transitions /I/. The noise is usually assumed to be a simple stochastic model process with definite properties which allow in some cases even exact solutions. The simplest continuous stochastic process is the Gaussian white noise (GWN) with autocorrelation W(t-t')= < ~t Et 'V =pa(t-t')' i.e. random events at different times are not correlated. The OrnsteinUhlenbeck-process (OUP) has an exponential decay of correlations W(t-t')=D/r exp(Jt-t'J/E ), i.e. a finite correlation time. The simplest discrete stochastic process is the dichotomous Markovian process (DMP) Jumping between two states (Fig. I). The autocorrelation is exponential as for the previous case.
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