Advances of the probability density evolution method for nonlinear stochastic systems

New advances of the probability density evolution method for nonlinear stochastic systems are presented. The principle of preservation of probability, as a fundamental law of stochastic systems, is firstly revisited. It provides a unified basis for the probability density evolution equations holding for different types of stochastic systems. By integrating the random event description of this principle into the uncoupled physical equation, the generalized density evolution equation (GDEE) is derived. Some new perspectives, including the property of independent evolution of partial probability density function and the paths of ensemble evolution and point evolution, are provided towards setting a solid foundation for the methods of partition of probability-assigned space and numerical discretization of the GDEE. On this basis, new advances and extensions are outlined in the aspects of numerical methods, an extension of the GDEE to generic stochastic systems and applications to fluctuation of nonlinear systems and stochastic optimal control of structures. Problems to be further explored are pointed out.

► The point evolution and ensemble evolution based on a new form of GDEE are discussed.
► The advance in numerical algorithm by employing Q-SPM points is exemplified.
► Extension of PDEM to generic–not limited to dynamical–systems is discussed.
► The fluctuation of the stochastic dynamical system is exposed via PDEM.
► Theoretical and experimental studies on stochastic optimal control are outlined.