On the relations between Markovian master equations and stochastic differential equations

We have investigated the relationship between Markovian master equations (m.e.) and the corresponding stochastic differential equations (s.d.e.) for closed systems, i.e., systems not subjected to external pumping. We show that the form of the fluctuations in the s.d.e., i.e., additive or multiplicative, depends upon the properties of the kernel of the m.e. and the range of the state space of the stochastic variable(s), i.e., bounded or unbounded. The knowledge of these two properties of the m.e. permits the determination of the way in which the fluctuations enter into the s.d.e. (i.e., additive or multiplicative) and the calculation of their statistics. Several examples are presented to illustrate the general theory.