Higher-Order Laplace Equations and Hyper-Cauchy Distributions

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• 作者：Enzo Orsingher ; Mirko D’Ovidio
• 关键词：Pseudo ; processes ; Stable processes ; Cauchy processes ; Higher ; order Laplace equations ; Airy functions ; Modified Bessel functions ; 60G52 ; 35C05
• 刊名：Journal of Theoretical Probability
• 出版年：2015
• 出版时间：March 2015
• 年：2015
• 卷：28
• 期：1
• 页码：92-118
• 全文大小：389 KB
• 参考文献：1. Beghin, L, Orsingher, E, Sakhno, L, Kozachenko, YV (2007) On the solutions of linear odd-order heat-type equations with random initial conditions. J. Stat. Phys. 127: pp. 721-739 CrossRef
  2. D’Ovidio, M., Orsingher, E.: Composition of processes and related partial differential equations. J. Theor. Probab. 24, 342-75 (2011)
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  6. Lachal, A (2003) Distributions of sojourn time, maximum and minimum for pseudo-processes governed by higher-order heat-type equations. Electron. J. Probab. 8: pp. 1-53
  7. Ladokhin, V.I.: On the “Measure-in Functional Space Corresponding to a Complex Diffusion Coefficient (Russian), vol. 123, pp. 36-2. Gos. Univ. U?en, Zap. (1963)
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  9. Nane, E (2008) Higher order pde’s and iterated processes. Trans. Am. Math. Soc. 360: pp. 2681-2692 CrossRef
  10. Orsingher, E.: Processes governed by signed measures connected with third-order “heat-type-equations. Litovsk. Mat. Sb. 31(2), 323-36 (1991). Translation in Lithuanian Math. J. 31(2), 220-31 (1992)
  11. Orsingher, E, Beghin, L (2009) Fractional diffusion equations and processes with randomly varying time. Ann. Probab. 37: pp. 206-249 CrossRef
  12. Orsingher, E, D’Ovidio, M (2011) Vibrations and fractional vibrations of rods, plates and Fresnel pseudo-processes. J. Stat. Phys. 145: pp. 143-174 CrossRef
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Probability Theory and Stochastic Processes
  Statistics
  
• 出版者：Springer Netherlands
• ISSN：1572-9230

文摘

In this paper we introduce new distributions which are solutions of higher-order Laplace equations. It is proved that their densities can be obtained by folding and symmetrizing Cauchy distributions. Another class of probability laws related to higher-order Laplace equations is obtained by composing pseudo-processes with positively skewed stable distributions which produce asymmetric Cauchy densities in the odd-order case. Special attention is devoted to the third-order Laplace equation where the connection between the Cauchy distribution and the Airy functions is obtained and analyzed.