ADVANCES IN THE RESEARCH ON PROBABILITY DENSITY EVOLUTION EQUATIONS OF STOCHASTIC DYNAMICAL SYSTEMS

Based on the ideas ofprobability density evolution, the history, development andapplications of the probability density evolution equations areelaborated in this paper. First, the physical meaning of theprinciple of preservation of probability is clarified, and theprinciple is then presented in terms of random event descriptionand state space description, respectively. Meanwhile, theintrinsic relationship between the probability density evolutionand the physical evolution of the system is elucidated, i.e. thephysical state evolution of the system is the inherent mechanismunderlying the probability density evolution.By incorporating the two descriptions of the principle ofpreservation of probability into the physical evolution equationsof the stochastic system, the classical probability density evolutionequations including the Liouville equation, FPK equation and theDostupov-Pugachev equation are revisited via methodologiesdifferent from the existing ones. The physical meaning of theseequations is clarified together with the reason why their dimensioncannot be reduced. Moreover, combining the random eventdescription of the principle of preservation of probability withthe uncoupled physical equation leads to the generalized densityevolution equation with its physical sense exposed. Theapplication of the probability density evolution theory isexemplified by the probability density evolution analysis of theresponse of nonlinear structures, and the problems in need offurther studies are pointed out at the end of paper.