Internal and external fluctuations a unified approach

It is a quite natural endeavour to search for a unified mathematical framework of describing internal and external fluctuations. Internal fluctuations used to be described by the Markovian master equation. Sancho San Miguel (1984) offered two equivalent techniques for a unified theory, at least for single-variable systems, when internal fluctuations were modelled specifically by a one-step Markovian master equation, and external noise was considered by dichotomous noise. 

The methods start from an evolution equation describing the internal fluctuation. In the next step a fixed value of an external parameter, namely the infinitesimal transition probability, is substituted by a stochastic process. This procedure can be done in the master equation or in the equation for the generating function. The first case leads to an integrodifferential equation, while the second model leads to a stochastic partial differential equation. 

Another technique was adopted by Horsthemke Lefever (1984a), to give a joint description of the two classes of fluctuations in order to predict the behaviour of systems of finite size. They started with a stochastic differential equation to describe internal fluctuations  

V is the volume of the system. To include finite correlation time the noise was described by the following Ornstein-Uhlenbeck process  

The influence of the external noise is taken into account as another Ornstein-Uhlenbeck process given by 

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