This chapter summarizes briefly the main tool of our dynamical analysis, that is, the basic dynamical equation for stochastic systems or models in this book.

It is known as the master equation in the literature of physics, and as the backward Chapman-Kolmogorov equation in the probability literature.

After decribing the transition rates that appear in the master equation, we explain, following van Kampen (1992, p. 253), apower-series expansionmethod of solving the master equation, by which we extract dynamical equations for the aggregate variables and derive equations for fluctuations about the locally stable equilibria. This equation, which governs probability distributions for the fluctuations about the equilibria, is known as the Fokker-Planck equation. These methods are illustrated and discussed further in Chapters 4 and 5. See Aoki (1996a, Secs. 5.2, 5.3). See also Risken (1989, p. 11) or Gardiner (1990, Chap. 7).

3.1