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Probability distributions for one component equations with multiplicative noise

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  • Deutsch, J.M.

Abstract

Systems described by equations involving both multiplicative and additive noise are common in nature. Examples include convection of a passive scalar field, polymers in turbulent flow, and noise in dye lasers. In this paper the one component version of this problem is studied. The steady state probability distribution is classified into two different types of behavior. One class has power law tails and the other is of the form of an exponential to a power law. The value of the power law exponent is determined analytically for models having colored gaussian noise. It is found to only depend on the power spectrum of the noise at zero frequency. When non-gaussian noise is considered it is shown that stretched exponential tails are possible. An intuitive understanding of the results is found and makes use of the Lyapunov exponents for these systems.

Suggested Citation

  • Deutsch, J.M., 1994. "Probability distributions for one component equations with multiplicative noise," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 208(3), pages 433-444.
    • Handle: RePEc:eee:phsmap:v:208:y:1994:i:3:p:433-444
    DOI: 10.1016/0378-4371(94)00055-7
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    Cited by:

    1. Andrew T. Balthrop, 2021. "Gibrat’s law in the trucking industry," Empirical Economics, Springer, vol. 61(1), pages 339-354, July.
    2. Aki-Hiro Sato & Hideki Takayasu, 2001. "Market price simulator based on analog electrical circuit," Papers cond-mat/0104318, arXiv.org.
    3. Lubashevsky, Ihor, 2013. "Equivalent continuous and discrete realizations of Lévy flights: A model of one-dimensional motion of an inertial particle," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(10), pages 2323-2346.
    4. Aki-Hiro Sato & Hideki Takayasu, 2001. "Derivation of ARCH(1) process from market price changes based on deterministic microscopic multi-agent," Papers cond-mat/0104313, arXiv.org.

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