## Cauchy problems of fractional order and stable processes.(English)Zbl 0718.35026

We study the Cauchy problem (of the fractional order) $(\partial /\partial t)^{\alpha}u(t,x)=(\partial /\partial x)^{\beta}u(t,x)\text{ for } 1\leq \alpha,\quad \beta \leq 2.$ After determining the pair of ($$\alpha$$,$$\beta$$) for which the solution exists uniquely, we investigate how the structure of the solution changes as ($$\alpha$$,$$\beta$$) changes. We also give the expression of the solution by the stable processes. In particular, for $$\beta =2$$, the positivity and the asymptotic behavior of the solution are studied.
Reviewer: Yasuhiro Fujita

### MSC:

 35G10 Initial value problems for linear higher-order PDEs 35B40 Asymptotic behavior of solutions to PDEs 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs

### Keywords:

Cauchy problem; fractional order; stable processes; positivity
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### References:

 [1] E. Artin, The Gamma Function. Holt, Rinehart and Winston, New York 1964. · Zbl 0144.06802 [2] P.L. Butzer and U. Westphal, An access to fractional differentiation via fractional difference quotients. Lecture Notes in Math. 457, Springer verlag. Berlin-Heidelberg-New York, 1975, 116–145. · Zbl 0307.26006 [3] H. Cartan, Elementary Theory of Analytic Functions of One or Several Complex Variables. Addison-Wesley Publ. Co. Inc., Reading, Mass., 1963. · Zbl 0121.30501 [4] K.L. Chung, A Course in Probability Theory. Academic Press, New York-San Francisco-London, 1974. · Zbl 0345.60003 [5] A. Friedman, Stochastic Differential Equations and Applications Vol. 1. Academic Press, New York-San Francisco-London, 1974. [6] B.V. Gnedenko and A.N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables (translated from the Russian). Addison-Wesley Publ. Co. Inc., Reading, Mass., 1954. · Zbl 0056.36001 [7] K. Ito, Lectures on Stochastic Processes. Tata Institute of fundamental Research, Bombay, 1961. [8] T. Kawata, Fourier Analysis in Probability Theory. Academic Press, New York-San Francisco-London, 1972. · Zbl 0271.60022 [9] E. Lukacs, Characteristic Functions. Griffin, London, 1960. · Zbl 0087.33605 [10] J.L. Mijnheer, Sample Path Properties of Stable Processes. Mathematical Central Tracts 59, Mathematisch Centrum Amsterdam, 1975. · Zbl 0307.60066 [11] S. Mizohata, The Theory of Partial Differential Equations. Cambridge University Press, 1973. · Zbl 0263.35001 [12] K.B. Oldham and J. Spanier, The Fractional Calculus. Academic Press, New York-San Francisco-London, 1974. · Zbl 0292.26011 [13] B. Ross, A brief history and exposition of the fundamental theory of fractional calculus. Lecture Notes in Math. 457, Springer-Verlag, Berlin-Heidelberg-New York, 1975, 1–36. [14] W.R. Schneider and W. Wyss, Fractional diffusion and wave equations. J. Math. Phys.,30 (1989), 134–144. · Zbl 0692.45004 [15] H. Takayasu,f power spectrum and stable distribution. J. Phys. Soc. Japan,56 (1987), 1257–1260. [16] H. Takayasu, Fractal clusters and stable distribution. J. Phys. Soc. Japan,57 (1988), 2585–2587.
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