Solution of generalized density evolution equation via a family of δsequences

Wenliang Fan, Jie Li

doi:10.6052/0459-1879-2009-3-2007-657

Volume 41 Issue 3

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Wenliang Fan, Jie Li. Solution of generalized density evolution equation via a family of δsequences[J]. Chinese Journal of Theoretical and Applied Mechanics, 2009, 41(3): 398-409. doi: 10.6052/0459-1879-2009-3-2007-657

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Wenliang Fan, Jie Li. Solution of generalized density evolution equation via a family of δsequences[J]. Chinese Journal of Theoretical and Applied Mechanics, 2009, 41(3): 398-409. doi: 10.6052/0459-1879-2009-3-2007-657

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Solution of generalized density evolution equation via a family of δsequences

doi: 10.6052/0459-1879-2009-3-2007-657

Wenliang Fan, Jie Li

Received Date: 2007-12-28
Rev Recd Date: 2008-09-11
Publish Date: 2009-05-18

Abstract

Abstract

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.
- Principle of Preservation of Probability,
- Generalized Density Evolution Equation,
- Formal Solution,
- δ Sequences,
- Non-Parameter Density Estimation