Article2020

Path Integral Methods for the Probabilistic Analysis of Nonlinear Systems Under a White-Noise Process

30 citations0/0 files storedASCE-ASME Journal of Risk and Uncertainty in Engineering Systems Part B Mechanical Engineering

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Probabilistic and Robust Engineering Design Probability and Statistical Research stochastic dynamics and bifurcation

Abstract

Abstract In this paper, the widely known path integral method, derived from the application of the Chapman–Kolmogorov equation, is described in details and discussed with reference to the main results available in literature in several decades of contributions. The most simple application of the method is related to the solution of Fokker–Planck type equations. In this paper, the solution in the presence of normal, α-stable, and Poissonian white noises is first discussed. Then, application to barrier problems, such as first passage problems and vibroimpact problems is described. Further, the extension of the path integral method to problems involving multi-degrees-of-freedom systems is analyzed. Lastly, an alternative approach to the path integration method, that is the Wiener Path integration (WPI), also based on the Chapman–Komogorov equation, is discussed. The main advantages and the drawbacks in using these two methods are deeply analyzed and the main results available in literature are highlighted.

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References

1976Probabilistic Theory of Structural Dynamics 1990Random vibration and statistical linearization 2009Stochastic Dynamics of Structures 1951On stochastic differential equations 2014Stochastic Differential Equations 1997A state-of-the-art report on computational stochastic mechanics 1951On a Formula Concerning Stochastic Differentials 2004Probability density evolution method for dynamic response analysis of structures with uncertain parameters

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