Solution of generalized density evolution equation via a family of δsequences

In the stochastic dynamics, it is one of the mostimportant purposes to acquire the probability density function and itsevolution process of the stochastic responses. The probability densityfunction of the responses or state vector of a stochastic dynamical systemis usually governed by some type of probability density evolution equationssuch as the Liouville, FPK or the Dostupov-Pugachev equation. However, theseequations in high dimension are too hard to obtain the solutions. Thegeneralized density evolution equation (GDEE), of which the dimension isindependent to the original dynamical system, provides a new possibility oftackling nonlinear stochastic systems. In this paper, based on the formalsolution of the GDEE, introducing the asymptotic sequences of the Diracδ function, a new numerical solution for the GDEE δ isproposed, to name as the Solution of GDEE via a family of δSequences. In addition, it is found that the non-parameter densityestimation can be regarded as a specific case of the proposed method. Atlast, the rationality and effectiveness of the proposed method is verifiedby some cases.