The principle of preservation of probability and the generalized density evolution equation

The present paper aims to provide a uniform and rigorous theoretical basis for the family of newly developed probability density evolution method. Conservation laws are among the most important features of continuum systems, so is the principle of preservation of probability for stochastic dynamical systems. The classical Liouville equation together with its Dostupov–Pugachev extension is firstly discussed. They could be reasonably thought to hold for stochastic systems where the randomness could be characterized by finite random variables but unfortunately they are unfeasible for practical applications because of analytical and numerical intractability. The generalized density evolution equation in conjunction with its numerical implementation procedure is then discussed with assistance of the formal solution. Comparing the Liouville equation and the generalized density evolution equation finds that the former is essentially based on the state space while the latter is on the ground of substantial particle description. The principle of preservation of probability is accordingly revisited from the two descriptions: the state space description and the random event description. On the clear basis, the generalized density evolution equation is derived once again in a more natural way. Underlying problems open for investigations and practical applications and possible extensions are outlined.