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Local nondeterminism and local times for stable processes. (English) Zbl 0659.60106
Our main theorem gives sufficient conditions for symmetric stable processes and fields to have a jointly continuous local time. The approach is through the $L\sp p$-representation for such processes. We develop a measure of dependence for vectors in a normed linear space and use that to analyze the probabilistic independence of the increments of a stable process. Local nondeterminism is defined for stable processes and shown to be equivalent to “locally approximately independent increments.” Sufficient conditions for several classes of stable processes to be local nondeterministic are given. These ideas are extended to multidimensional stable random fields and we prove existence of jointly continuous local times. The results extend most Gaussian results to their stable analogs.
Reviewer: J.P.Nolan

60J55Local time, additive functionals
60G17Sample path properties
60G60Random fields
Full Text: DOI
[1] Adler, R.J.: Geometry of random fields. New York: Wiley 1981 · Zbl 0478.60059
[2] Berman, S.M.: Local nondeterminism and local times of Gaussian processes. Indiana Univ. Math. J. 23, 69--94 (1973) · Zbl 0264.60024 · doi:10.1512/iumj.1973.23.23006
[3] Blumenthal, R.M., Getoor, R.K.: Markov processes and potential theory. New York: Academic Press 1968 · Zbl 0169.49204
[4] Cambanis, S., Miller, G.: Linear problems in path order and stable processes. SIAM J. Appl. Math. 41, 43--69 (1983) · Zbl 0466.60044 · doi:10.1137/0141005
[5] Cuzick, J., DuPreez, J.P.: Joint continuity of Gaussian local times. Ann. Probab. 10, 810--817 (1982) · Zbl 0492.60032 · doi:10.1214/aop/1176993789
[6] Davis, P.J.: Interpolation and approximation. New York: Blaisdell 1963 · Zbl 0111.06003
[7] Ehm, W.: Sample function properties of multiparameter stable processes. Zeit. Wahrscheinlichkeitstheor. Verw. Geb. 56, 195--228 (1981) · Zbl 0471.60046 · doi:10.1007/BF00535741
[8] Geman, D., Horowitz, J.: Occupation densities. Ann. Probab. 8, 1--67 (1980) · Zbl 0499.60081 · doi:10.1214/aop/1176994824
[9] Hardin, C.: On the spectral representation of symmetric stable processes. J. Multivariate Anal. 12, 385--401 (1982) · Zbl 0493.60046 · doi:10.1016/0047-259X(82)90073-2
[10] Ito, K., McLean, H.P.: Diffusion processes and their sample paths. New York: Academic Press 1965 · Zbl 0127.09503
[11] Levy, P.: Processes stochastiques et mouvement brownien 2nd ed. Paris: Gauthier-Villars, 1965
[12] Marcus, M.B., Pisier, G.: Characterizations of almost surely continuous p-stable random Fourier series and strongly stationary processes. Acta Math. 152, 245--301 (1984) · Zbl 0547.60047 · doi:10.1007/BF02392199
[13] Nolan, J.P.: Local times for stable processes. Ph.D. Dissertation, Univ. of Virginia, 1982
[14] Nolan, J.P., Sinkala, Z.: Equivalent norms on analytic subspaces of L p . J. Math. Anal. Appl. 126, 238--249 (1987) · Zbl 0626.46017 · doi:10.1016/0022-247X(87)90089-8
[15] Nolan, J.P.: Path properties of index-{$\beta$} stable fields. Ann. Probab. 16, 1596--1607 (1988) · Zbl 0673.60043 · doi:10.1214/aop/1176991586
[16] Pitt, L.D.: Local times for Gaussian vector fields. Indiana Univ. Math. J. 27, 309--330 (1978) · Zbl 0382.60055 · doi:10.1512/iumj.1978.27.27024
[17] Rosinski, J.: On stochastic integral representation of stable processes with sample paths in Banach spaces. J. Multvarate Anal. 20, 277--302 (1986) · Zbl 0606.60041 · doi:10.1016/0047-259X(86)90084-9
[18] Singer, I.: Bases in banach spaces I. New York Berlin Heidelberg: Springer 1970 · Zbl 0198.16601
[19] Trotter, H.F.: A property of Brownian motion paths. Illinois J. Math. 2, 425--432 (1958) · Zbl 0117.35502