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

Based on the ideas of probability density evolution,the history,development and applications of the probability density evolution equations are elaborated in this paper.First,the physical meaning of the principle of preservation of probability is clarified,and the principle is then presented in terms of random event description and state space description,respectively.Meanwhile,the intrinsic relationship between the probability density evolution and the physical evolution of the system is elucidated,i.e.the physical state evolution of the system is the inherent mechanism underlying the probability density evolution. By incorporating the two descriptions of the principle of preservation of probability into the physical evolution equations of the stochastic system,the classical probability density evolution equations including the Liouville equation,FPK equation and the Dostupov-Pugachev equation are revisited via methodologies different from the existing ones.The physical meaning of these equations is clarified together with the reason why their dimension cannot be reduced.Moreover,combining the random event description of the principle of preservation of probability with the uncoupled physical equation leads to the generalized density evolution equation with its physical sense exposed.The application of the probability density evolution theory is exemplified by the probability density evolution analysis of the response of nonlinear structures,and the problems in need of further studies are pointed out at the end of paper.