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Research Articles

Stable distributions and pseudo-processes related to fractional Airy functions

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Pages 435-450 | Received 07 Feb 2023, Accepted 16 Oct 2023, Published online: 29 Oct 2023

References

  • Bernstein, F. (1918). Über das Fourierintegral ∫0∞e−x4cos⁡txdx. Math. Ann. 79(3): 265–268. doi:10.1007/BF01458208.
  • Burwell, W. R. (1924). Asymptotic expansions of generalized hyper-geometric functions. Proc. London Math. Soc. s2-22(1): 57–72. doi:10.1112/plms/s2-22.1.57.
  • Krylov, V. Y. (1960). Some properties of the distribution corresponding to equation ∂u∂t=(−1)q+1∂2qu∂x2q. Sov. Math. Dokl. 1:760–763.
  • Miyamoto, M. (1966). An extension of certain quasi-measure. Proc. Jpn. Acad. 42(2): 70–74.
  • Hochberg, K. J. (1978). A signed measure on path space related to Wiener measure. Ann. Probab. 6(3): 433–458.
  • Ladokhin, V. I. (1962). On non-positive distributions. Kazan. Gos. Univ. Uĉen. Zap. 122(4): 53–64.
  • Daletskii, Y. L., Fomin, S. V. (1965). Generalized measures in function spaces. Theory Probab. Appl. 10(2): 304–316. doi:10.1137/1110035.
  • Debbi, L. (2006). Explicit solutions of some fractional partial differential equations via stable subordinators. J. Appl. Math. Stoch. Anal. 5: 093502. doi:10.1155/JAMSA/2006/93502.
  • Debbi, L. (2007). On some properties of a higher order fractional differential operator which is not in general selfadjoint. Appl. Math. Sci. 1(27): 1325–1339.
  • Orsingher, E. (1992). Processes governed by signed measures connected with third-order?heat-type? equations. Lith. Math. J. 31(2): 220–231. doi:10.1007/BF00970819.
  • Lachal, A. (2003). Distributions of Sojourn time, maximum and minimum for pseudo-processes governed by higher-order heat-type equations. Electr. J. Probab. 8: 1–53.
  • Nakajima, T., Sato, S. (2014). An approach to the pseudoprocess driven by the equation ∂∂t=−A∂3∂x3 by a random walk. Kyoto J. Math.. 54(3): 507–528.
  • Lachal, A. (2007). First hitting time and place, monopoles and multipoles for pseudo-processes driven by the equation ∂u/∂t=±∂Nu/∂xN. Electr. J. Probab. 12: 300–353.
  • Nishioka, K. (1997). The first hitting time and place of a half-line by a biharmonic pseudo process. Jpn. J. Math. 23(2): 235–280. doi:10.4099/math1924.23.235.
  • Nishioka, K. (2001). Boundary conditions for one-dimensional biharmonic pseudo process. Electr. J. Probab. 6: 1–27.
  • Bonaccorsi, S., Mazzucchi, S. (2015). High order heat-type equations and random walks on the complex plane. Stoch. Process. Appl. 125(2): 797–818. doi:10.1016/j.spa.2014.08.010.
  • Lachal, A. (2014). From pseudorandom walk to pseudo-Brownian motion: First exit time from a one-sided or a two-sided interval. Int. J. Stoch. Anal. 2014: 1–49. doi:10.1155/2014/520136.
  • Orsingher, E., Toaldo, B. (2014). Pseudoprocesses related to space-fractional higher-order heat-type equations. Stoch. Anal. Appl. 32(4): 619–641. doi:10.1080/07362994.2014.911107.
  • Orsingher, E., Ovidio, M. (2012). Probabilistic representation of fundamental solutions to ∂u∂t=κm∂mu∂xm. Electr. Commun. Probab. 17: 1–12.
  • Askari, H., Ansari, A. (2020). On Mellin transforms of solutions of differential equation χ(n)(x)+γnxχ(x)=0. Anal. Math. Phys. 10(4): 57.
  • Zolotarev, V. M. (1986). One-Dimensional Stable Distributions. Providence: American Mathematical Society.
  • Ansari, A., Askari, H. (2014). On fractional calculus of A2n+1(x) function. Appl. Math. Comput. 232: 487–497. doi:10.1016/j.amc.2014.01.058.
  • Watson, G. N., (1951). A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge: Cambridge University Press.
  • Prudnikov, A., Brychkov, Y., Marichev, O. (1989). More Special Functions. Vol. 3, In Integrals and Series. New York: Gordon and Breach.
  • Feller, W., (1952). On a generalization of Marcel Riesz’ Potentials and the Semi-Groups generated by them. In Meddelanden Lunds Universitetes Matematiska Seminarium (Comm. Sém. Mathém. Université de Lund), Tome Suppl. dédié a M. Riesz, 73–81.
  • Mainardi, F., Luchko, Y., Pagnini, G. (2007). The fundamental solution of the space-time fractional diffusion equation. Fract. Calcul. Appl. Anal. 4.
  • Zorich, V. A. (2004). Mathematical Analysis I. Berlin: Springer.

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