Stochastic differential equations some concepts and comments

2 Stochastic differential equations some concepts and comments Let us consider a stochastic dynamic system  

In many applications of the theory of noise-induced transition it is assumed that fo is linear in A, i.e. 

Stochastic dynamic systems can be classified according to the very nature of/. Arnold Kliemann (1981) summarised the qualitative behaviour of x both for linear and nonlinear systems (for a condensed survey see Arnold (1981). The term linear is not specific here, since / can be linear either in state or in noise, even in both. In applications it is assumed very often that the forcing function has a systematic or deterministic part, and a term due to the rapidly varying, highly irregular random effects  

This is a verbal version of the old Langevin equation (Langevin, 1908). According to a standard further assumption the random term is a linear function of a white noise . White noise is considered as a stationary Gaussian process with = 0 and ] = 8 / - t, where 5 is the 

It is well-known today that the heuristic derivation of the Langevin equation is mathematically not well-founded. A stochastic process X, obeys an Ito stochastic differential equation (SDE) which has the form (in the autonomous case). 

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