A note on the principle of preservation of probability and probability density evolution equation

Probabilistic Engineering Mechanics

Volumn 24, Issue 1, 2009, Pages 51-59

  Chen, Jian Bing ^a   Li, Jie ^a  

^a TONGJI UNIVERSITY   (China)

Author keywords

Fokker Planck equation; Generalized density evolution equation; Principle of preservation of probability; Probability density evolution equation; Random event description; State space description

Indexed keywords

CLARIFICATION; DECODING; DIFFERENTIAL EQUATIONS; DYNAMICAL SYSTEMS; EQUATIONS OF MOTION; FOKKER PLANCK EQUATION; LAGRANGE MULTIPLIERS; PROBABILITY; PROBABILITY DENSITY FUNCTION; RECONNAISSANCE AIRCRAFT; SCHRODINGER EQUATION;

GENERALIZED DENSITY EVOLUTION EQUATION; PRINCIPLE OF PRESERVATION OF PROBABILITY; PROBABILITY DENSITY EVOLUTION EQUATION; RANDOM EVENT DESCRIPTION; STATE SPACE DESCRIPTION;

RANDOM PROCESSES;

EID: 56449130667     PISSN: 02668920     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.probengmech.2008.01.004     Document Type: Article

References (28)

1. Bontempi F., and Faravelli L. Lagrangian/Eulerian description of dynamic system. Journal of Engineering Mechanics 124 8 (1998) 901-911
2. Caughey T.K. Nonlinear theory of random vibrations. Advances in applied mechanics vol. 11 (1971), Academic Press, New York
3. Chen J.B., and Li J. Dynamic response and reliability analysis of nonlinear stochastic structures. Probabilistic Engineering Mechanics 20 1 (2005) 33-44
4. Dafermos C.M. Hyperbolic conservation laws in continuum physics (2000), Springer-Verlag, Berlin
5. Dostupov B.G., and Pugachev V.S. The equation for the integral of a system of ordinary differential equations containing random parameters. Automatika i Telemekhanika 18 (1957) 620-630
6. Einstein A. Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Annalen der Physik (Leipzig) 17 (1905) 549-560
7. Frank T.D. Nonlinear Fokker-Planck equations: Fundamentals and applications (2005), Springer
8. Fung Y.C. A first course in continuum mechanics (1994), Prentice-Hall, Inc.
9. Fokker A.D. Die mittlere Energie rotierender elektrischer Dipole im Strahlungsfeld. Annalen der Physik (Leipzig) 43 (1914) 810-820
10. Gardiner C.W. Handbook of stochastic methods for physics, chemistry and the natural sciences. 2nd ed. (1983), Springer, Berlin
11. Gibbs J.W. Elementary principles in statistical mechanics (1902) Reprinted by Ox Bow Press, 1981
12. Kleinert H. Path integrals (1995), World Scientific Publishing Co. Pte. Ltd, Singapore
13. Kolmogoroff A.N. Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung. Mathematische Annalen 104 (1931) 415-458
14. Li J., and Chen J.B. The number theoretical method in response analysis of nonlinear stochastic structures. Computational Mechanics 39 6 (2007) 693-708
15. Li J., and Chen J.B. The principle of preservation of probability and the generalized density evolution equation. Structural Safety 30 (2008) 65-77
16. Li J., Chen J.B., and Fan W.L. The equivalent extreme-value event and evaluation of the structural system reliability. Structural Safety 29 (2007) 112-131
17. Lin Y.K. Probabilistic theory of structural dynamics (1967), McGraw-Hill Book Company, New York
18. Loeve M. Probability Theory (1977), Springer-Verlag, Berlin
19. Lützen J. Joseph Liouville 1809-1882: Master of pure and applied mathematics (1990), Springer-Verlag, New York
20. Naess A., and Johnsen J.M. Response statistics of nonlinear, compliant offshore structures by the path integral solution method. Probabilistic Engineering Mechanics 8 2 (1993) 91-106
21. Naess A., and Moe V. Efficient path integral methods for nonlinear dynamics systems. Probabilistic Engineering Mechanics 15 2 (2000) 221-231
22. Øksendal B. Stochastic differential equations: An introduction with applications. 6th ed. (2005), Springer-Verlag, Berlin
23. Planck M. Ueber einen Satz der statistichen Dynamik und eine Erweiterung in der Quantumtheorie. Sitzungberichte der Preussischen Akadademie der Wissenschaften (1917) 324-341
24. Risken H. The Fokker-Planck equation: Methods of solution and applications (1984), Springer-Verlag, Berlin
25. In: Schuëller G.I. (Ed). A state-of-the-art report on computational stochastic mechanics. Probabilistic Engineering Mechanics 12 4 (1997) 197-321
26. Soong T.T. Random differential equations in science and engineering (1973), Academic Press, New York (London)
27. Syski R. Stochastic differential equations. In: Saaty T.L. (Ed). Modern nonlinear equations (1967), McGraw-Hill, New York [Chapter 8]
28. Zhu W.Q. Nonlinear stochastic dynamics and control in Hamiltonian formulation. Applied Mechanics Reviews 59 (2006) 230-248