Abstract
In this paper odd-order heat-type equations with different random initial conditions are examined. In particular, we give rigorous conditions for the existence of the solutions in the case where the initial condition is represented by a strictly ϕ –subGaussian harmonizable process η = η (x). Also the case where η is represented by a stochastic integral with respect to a process with independent increment is studied.
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L. Beghin, K. Hochberg and E. Orsingher, Stoch. Process. Appl. 85:209–223 (2000).
L. Beghin, V. P. Knopova, N. N. Leonenko and E. Orsingher, J. Stat. Phys. 99(3/4):769–781 (2000).
L. Beghin and E. Orsingher, Stoch. Process. Appl. 115:1017–1040 (2005).
L. Beghin, E. Orsingher, and T. Ragozina, Stoch. Process. Appl. 94:71–93 (2001).
V. V. Buldygin and Yu. V. Kozachenko, Metric Characterization of Random Variables and Random Processes. Translations of Mathematical Monographs. 188 (AMS, American Mathematical Society, Providence, RI, 2000), 257 p.
J. L. Doob, Trans. Am. Math. Soc. 42:107–140 (1937).
C. W. Gardiner, Handbook of Stochastic Methods (Springer, Berlin, 1985).
R. Giuliano Antonini, Yu. V. Kozachenko, and T. Nikitina, Rendiconti Accademia Nazionale delle Scienze XL. Memorie di Matematica e Applicazioni 121. XXVII:95–124 (2003).
K. J. Hochberg and E. Orsingher, Stoch. Proc. Appl. 52:273–292 (1994).
G. A. Hunt, Trans. Am. Math. Soc. 71:38–69 (1951).
J. K. Hunter and J. Scheurle, Physica D 32:253–268 (1988).
T. Iizuka, Phys. Lett. A 181:39–42 (1993).
T. Kakutani and H. Ono, J. Phys. Soc. Jpn. 26:1305–1318 (1969).
T. Kawahara, J. Phys. Soc. Jpn. 33:260–264 (1972).
V. P. Knopova, Random Oper. Stoch. Equ. 12(1):35–42 (2004).
Yu. V. Kozachenko, Dokl. Akad. Nauk Ukr. SSR, Ser. A. 9:14–16 (1984).
Yu. V. Kozachenko, Theory Probab. Math. Stat. 40:43–50 (1990).
Yu. V. Kozachenko and E. I. Ostrovskij, Theory Probab. Math. Stat. 32:45–56 (1986).
Yu. V. Kozachenko and I. Rozora, Random Oper. Stoch. Equ. 3:275–296 (2003).
Yu. V. Kozachenko and G. I. Slivka, Theor. Probab. Math. Statist. 69:67–83 (2004).
A. Lachal, Electron. J. Prob. 8(20):1–53 (2003).
A. Lachal, First hitting time and place, monopoles and multipoles for pseudo-processes driven by the equation \(\frac{\partial u}{\partial t}=\pm \frac{\partial ^{N}u}{\partial x^{N}}\), preprint (2006).
M. J. Lighthill, Waves in Fluids (Cambridge University Press, Cambridge, 1978).
M. Loève, Probability Theory. Foundations. Random sequences. (D. van. Nostrand Co., Inc. XV., New York, 1955) 515 p.
T. Mikosch and R. Norvaiša, Bernoulli 6(3):401–434 (2000).
H. Nagashima, Phys. Lett. A 105:439–442 (1984).
H. Nagashima and M. Kuwahara, J. Phys. Soc. Jpn. 105:439–442 (1981).
R. Norvaiša and G. Samorodnitsky, Ann. Probab. 22(4):1904–1929 (1994).
E. Orsingher, Lith. Math. J. 31:321–334 (1991).
B. S. Rajput and J. Rosinski, Probab. Theory Rel. Fields 82:451–487 (1989).
F. Tappert, Nonlinear Wave Motion, Lecture Notes in Applied Math. (AMS) vol. 15, pp. 215–216 (1974).
M. Wadati, J. Phys. Soc. Jpn. 52:2642–2648 (1983).
G. B. Whitham, Linear nad Nonlinear Waves. (John Wiley & Sons Inc., New York). Reprint of the 1974 original, A Wiley-Interscience Publ. (1999).
K. Yoshimura and S. Watanabe, J. Phys. Soc. Jpn. 51:3028–3055 (1982).
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Partially supported by the NATO grant PST.CLG.980408.
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Beghin, L., Kozachenko, Y., Orsingher, E. et al. On the Solutions of Linear Odd-Order Heat-Type Equations with Random Initial Conditions. J Stat Phys 127, 721–739 (2007). https://doi.org/10.1007/s10955-007-9309-x
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DOI: https://doi.org/10.1007/s10955-007-9309-x