On the distribution of a rotationally invariant α-stable process at the hitting time of a given hyperplane

Osypchuk, MM, Portenko, MI
Dopov. Nac. akad. nauk Ukr. 2018, 12:14-20
https://doi.org/10.15407/dopovidi2018.12.014
Section: Mathematics
Language: English
Abstract: 
We find out an explicit formula for the distribution of a rotationally invariant α-stable process at that moment of time, when it hits a given hyperplane for the first time. The case of $1 < \alpha \leqslant 2$ is considered.
Keywords: hitting time, local time, resolvent kernel, α-stable process
References: 
  1. Dynkin, E. B. (1965). Markov Processes. (Vols. 1, 2). New York: Academic Press; Berlin: Springer. doi: https://doi.org/10.1007/978-3-662-00031-1
  2. Blumenthal, R. M., Getoor, R. K. & Ray, D. B. (1961). On the distribution of first hits for the symmetric stable process. Trans. Am. Math. Soc., 99, No. 3, pp. 540-554.
  3. Port, S. C. (1967). Hitting times and potentials for recurrent stable processes. J. Anal. Math., 20, pp. 371-395. doi: https://doi.org/10.1007/BF02786681
  4. Rogozin, B. A. (1971). The distribution of the first ladder moment and height and fluctuation of a random walk. Theory Probab. Appl., 16, pp. 575-595. doi: https://doi.org/10.1137/1116067
  5. Caballero, M. E., Pardo, J. C. & Pérez, J. L. (2011). Explicit identities for Lévy process associated to symmetric stable processes. Bernoulli, 17, pp. 34-59. doi: https://doi.org/10.3150/10-BEJ275
  6. Kyprianou, A. E., Pardo, J. C. & Watson, A. R. (2014). Hitting distributions of α-stable processes via path censoring and self-similarity. Ann. Probab., 42, No. 1, pp. 398-430. doi: https://doi.org/10.1214/12-AOP790
  7. Bochner, S. (1959). Lectures on Fourier integrals. Princeton, New Jersey: Princeton Univ. Press.
  8. Osypchuk, M. M. & Portenko, M. I. (2016). On single-layer potentials for one class of pseudo-differential equations. Ukr. Math. J., 67, No. 11, pp. 1704-1720. doi: https://doi.org/10.1007/s11253-016-1184-7
  9. Blumenthal, R. M & Getoor, R. K. (1960). Some theorems on stable processes. Trans. Am. Math. Soc., 93, No. 2, pp. 263-273. doi: https://doi.org/10.1090/S0002-9947-1960-0119247-6
  10. Eidelman, S. D., Ivasyshen, S. D. & Kochubei, A. N. (2004). Analytic methods in the theory of differential and pseudo-differential equations of parabolic type. Operator Theory: Advances and Applications. (Vol. 152). Basel: Birkhäuser. doi: https://doi.org/10.1007/978-3-0348-7844-9
  11. Osypchuk, M. M. & Portenko, M. I. (2017). On constructing some membranes for a symmetric α-stable process. Commun. Stoch. Anal., 11, No. 1, pp. 11-20. doi: https://doi.org/10.31390/cosa.11.1.02
  12. Bateman, H. & Erdélyi, A. (1953). Higher transcendental functions. (Vol. 2). New York: McGraw-Hill.