A note on the principle of preservation of probability and probability density evolution equation

The present paper aims at clarifying the physical sense of the principle of preservation of probability. Using this principle as a unified fundamental, the probability density evolution equations, including the Liouville, Fokker–Planck and the Dostupov–Pugachev equation, are derived from the physical point of view. Further, it is pointed out that there exist different descriptions of this principle and, from these different descriptions, combining with the Eulerian or Lagrangian description of the associated dynamical system will lead to different probability density evolution equations. Particularly, when both the principle of preservation of probability and the motion of the dynamical systems are viewed from the Lagrangian description, we are led to the generalized probability density evolution equation. In contrast to the state space description, where the transition of probability is treated in different ways based on their different phenomenological origins, the essential point of the random event description is to view the transition of probability in a unified way because they result from the same source of random events.