×

zbMATH — the first resource for mathematics

A local time approach to the self-intersections of Brownian paths in space. (English) Zbl 0534.60070
Summary: We study the Brownian functional \(\alpha(x,B)=\iint_{B}\delta_ x(W_ t-W_ s)dsdt,\) where \(W_ t\) is a Brownian path in two or three dimensions. For B off the diagonal we identify \(\alpha\) (x,B) with a local time, and establish the Hölder continuity of \(\alpha\) (x,B) in both x and B.

MSC:
60J65 Brownian motion
60G17 Sample path properties
60J55 Local time and additive functionals
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Adler, R. (1978): The uniform dimension of the level sets of a Brownian sheet. Ann. Prob.6, 509-515 · Zbl 0378.60028 · doi:10.1214/aop/1176995535
[2] Berman, S. (1973): Local nondeterminism and local times of Gaussian processes. Ind. Univ. Math. J.23, 69-94 · Zbl 0264.60024 · doi:10.1512/iumj.1973.23.23006
[3] Cuzick, J.: Continuity of Gaussian local times, Ann. Prob. (to appear) · Zbl 0492.60033
[4] Dvoretzky, A., Erd?s, P., Katutani, S. (1950): Double points of paths of Brownian motion in n-space. Acta Sci. Math. Szeged, 12B, 75-81 · Zbl 0036.09001
[5] Edwards, S. (1965): The statistical mechanics of polymers with excluded volume. Proc. Phys. Sci.85, 613-624 · Zbl 0125.23205 · doi:10.1088/0370-1328/85/4/301
[6] Fristedt, B. (1967): An extension of a theorem of S. J. Taylor concerning the multiple points of the symmetric processes. Z. Wahrsch. Verw. Gebiete9, 62-64 · Zbl 0314.60030 · doi:10.1007/BF00535468
[7] Geman, D., Horowitz, J. (1980): Occupation densities, Ann. Probability8, 1-67 · Zbl 0499.60081 · doi:10.1214/aop/1176994824
[8] German, D., Horowitz, J., Rosen, J.: The local time of intersections for Brownian paths in the plane. Ann. Prob. (to appear).
[9] It?, K., McKean, H. P. (1965): Diffusion processes and their sample paths. New York: Academic Press · Zbl 0127.09503
[10] Kaufman, R. (1969): Une propri?t? m?trique des mouvement brownien. C. R. Acad. Sc. Paris, t.268, S?rie A, 727-728 · Zbl 0174.21401
[11] Kono, N. (1977): H?lder conditions for the local times of certain Gaussian processes with stationary increments. Proc. Jpn. Acad.53, Ser. A, No. 3, 84-87 · Zbl 0437.60057
[12] Pitt, L. (1978): Local times for Gaussian vector fields. Ind. Univ. Math. J.27, 309-330 · Zbl 0382.60055 · doi:10.1512/iumj.1978.27.27024
[13] Rosen, J. (1981): Joint continuity of the local time for the N-parameter Wiener process inR d . University of Massachusetts preprint
[14] Symanzik, K. (1969): Euclidean quantum field theory. In: Local Quantum Theory, Jost, R. (ed.) New York: Academic Press
[15] Taylor, S. J. (1966): Multiple points for the sample paths of the symmetric stable processes. Z. Wahrsch. Verw. Gebiete5, 247-264 · Zbl 0146.37905 · doi:10.1007/BF00533062
[16] Taylor, S. J. (1973): Sample path properties of processes with stationary independent increments. In: Stochastic Analysis, Kendall, D., Harding, E. (eds.) London: J. Wiley and Sons
[17] Tran, L. T. (1976): On a problem posed by Orey and Pruitt related to the range of the N-parameter Wiener process in Rd. Z. Wahrsch. Verw. Gebiete6, 170-180
[18] Trotter, H. (1958): A property of Brownian motion paths. Ill. J. Math.2, 425-432 · Zbl 0117.35502
[19] Westwater, J.: On Edwards’ model for long polymer chains. Univ. of Washington preprint · Zbl 0431.60100
[20] Wolpert, R. (1978): Wiener path intersections and local time. J. Funct. Anal.30, 329-340 · Zbl 0403.60069 · doi:10.1016/0022-1236(78)90061-7
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.