Abstract
Fix two rectangles A,B in [0,1]2. Then the size of the random set of double points of the Brownian sheet (W t )t∈[0,1]2 in ℝd, i. e. the set of pairs (s,t),where s ∈ A, t ∈ B, and W s = W t , can be measured as usual by a self-intersection local time. If A = B, we show that the critical dimension below which self-intersection local time exists, is given by d = 4. If A ⋂ B consists of an axial parallel line, it is 6, if it consists of a point or is empty, 8. In all cases, we derive the rate of explosion of canonical approximations of self-intersection local time for dimensions above the critical one, and determine its smoothness in terms of the canonical Dirichlet structure on Wiener space.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
R. Bass and D. Koshnevisan, Intersection local times and Tanaka formulas, Preprint, Univ. of Washington, 1992.
N. Bouleau and F. Hirsch, Dirichlet forms and analysis on Wiener space, W. de Gruyter, Berlin, 1991.
E. Dynkin, Self-intersection gauge for random walks and for Brownian motion, Ann. Probab. 16 (1988), 1–57.
P. Imkeller, V. Perez-Abreu, and J. Vives, Chaos expansions of double intersection local time of Brownian motion in ?d and renormalization (to appear in: Stoch. Proc. Appl.).
P. Imkeller and F. Weisz, The asymptotic behaviour of local times and occupation integrals of the N-parameter Wiener process in ?d Probab. Th. Rel. Fields 98 (1994), 47–75.
H. H. Kuo, Donsker’s delta function as a generalized Brownian functional and its application, Lecture Notes in Control and Information Sciences, vol. 49, Springer-Verlag, Berlin, 1983, pp. 167–178.
J. F. Le Gall, Sur le temps local d’intersection du mouvement brownien plan et la méthode de renormalisation de Varadhan, Sém. de Prob. XIX 1983/84. LNM 1123, Springer-Verlag, Berlin, 1985, pp. 314–331.
D. Nualart and J. Vives, Chaos expansion and local times, Publications Matematiques 36, 2 (1992), 827–836.
J. Rosen,A local time approach to the self-intersections of Brownian paths in space, Comm. Math. Phys. 88 (1983), 327–338.
J. Rosen, Stochastic integrals and intersections of Brownian sheets,Preprint, Univ. of Massachusetts.
J. Rosen, Self-intersections of random fields, Ann. Probab. 12 (1984), 108–119.
J. Rosen, Tanaka’s formula and renormalization for intersections of planar Brownian motion, Ann. Probab. 14 (1986), 1245–1251.
J. Rosen and M. Yor, Tanaka formulae and renormalization for triple intersections of Brown ian motion in the plane,Preprint.
N. R. Shieh, White noise analysis and Tanaka formula for intersections of planar Brownian motion, Nagoya Math. J. 122 (1991), 1–17.
K. Szymanzik, Euclidean quantum field theory, In: Local Quantum Theory ( R. Jost, ed.), Academic Press, New York, 1969.
S. R. S. Varadhan, Appendix to “Euclidean quantum field theory” by K. Szymanzik, In: Local Quantum Theory ( R. Jost, ed.), Academic Press, New York, 1969.
H. Watanabe, The local time of self-intersections of Brownian motions as generalized Brownian functionals, Letters in Math. Physics 23 (1991), 1–9.
S. Watanabe, Lectures on stochastic differential equations and Malliavin calculus, Springer-Verlag, Berlin, 1984.
R. Wolpert, Wiener path intersection and local time,J. Func. Anal. 30 (1978), 329–340.
M. Yor, Sur la représentation comme intégrales stochastiques des temps d’occupation du mouvement Brownian dans ?d, Sém. de Prob. XX. 1984/85. LNM 1204, Springer-Verlag,Berlin, 1986, pp. 543–552.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Springer Basel AG
About this paper
Cite this paper
Imkeller, P., Weisz, F. (1995). Critical Dimensions for the Existence of Self-Intersection Local Times of the Brownian Sheet in ℝd . In: Bolthausen, E., Dozzi, M., Russo, F. (eds) Seminar on Stochastic Analysis, Random Fields and Applications. Progress in Probability, vol 36. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7026-9_11
Download citation
DOI: https://doi.org/10.1007/978-3-0348-7026-9_11
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-7028-3
Online ISBN: 978-3-0348-7026-9
eBook Packages: Springer Book Archive