×

zbMATH — the first resource for mathematics

\(\alpha\) -self-similar Markov processes. (English) Zbl 0561.60085
Let ((X(t)), \(P^ x)\) be an \(\alpha\)-self-similar isotropic Markov process on \(R^ d\setminus \{0\}\). A representation of (X(t)), in terms of the radial and angular process which generalizes the skew product representation for Brownian motion is given.

MSC:
60J65 Brownian motion
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bingham, N.H.: Random walk on spheres. Z. Wahrsheinlichkeitstheorie. verw. Geb. 22, 169-192 (1972) · Zbl 0222.60043 · doi:10.1007/BF00536088
[2] Blumenthal, R.M., Getoor, R.K.: Markov processes and potential theory. New York: Academic Press 1968 · Zbl 0169.49204
[3] Bochner, S.: Positive zonal functions on spheres, Proc. Natl. Acad. Sci. (USA) 40, 1141-1147, (1954) · Zbl 0058.29101 · doi:10.1073/pnas.40.12.1141
[4] Bochner, S.: Harmonic analysis and the theory of probability. Univ. Calif. Press 1955 · Zbl 0068.11702
[5] Galmarino, A.R.: Representation of an isotropic diffusion as a skew product. Z. Wahrscheinlichkeitstheor. verw. Geb. 1, 359-378 (1963) · Zbl 0109.36303 · doi:10.1007/BF00533411
[6] Gangolli, R.: Isotropic infinitely-divisible measures on symmetric spaces. Acta Math. 111, 213-246 (1964) · Zbl 0154.43804 · doi:10.1007/BF02391013
[7] ItĂ´, K., McKean, H.P.: Diffusion processes and their sample paths. Berlin-Heidelberg-New York: Springer 1965 · Zbl 0127.09503
[8] Kiu, S.W.: Semi-stable Markov processes in R n. Stochastic Processes Appl. 10, 183-191 (1980) · Zbl 0436.60054 · doi:10.1016/0304-4149(80)90020-4
[9] Lamperti, J.W.: Semi-stable Markov processes I. Z. Wahrscheinlichkeitstheor. verw. Geb. 22, 205-225 (1972) · Zbl 0274.60052 · doi:10.1007/BF00536091
[10] Lamperti, J.W.: Semi-stable stochastic processes. Trans. Am. Math. Soc. 104, 62-78 (1962) · Zbl 0286.60017 · doi:10.1090/S0002-9947-1962-0138128-7
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.