## Chaos expansions of double intersection local time of Brownian motion in $$\mathbb{R}^ d$$ and renormalization.(English)Zbl 0822.60048

Let $$W = \{W_ t,\;t \in [0,1]\}$$ be a $$d$$-dimensional Brownian motion. Denoting by $$p^ d_ \varepsilon$$ the probability density of $$W_ \varepsilon$$, the authors derive the representation of $\alpha_ \varepsilon (x) = \int^ 1_ 0 \int^ 1_ 0 p^ d_ \varepsilon (W_ t - W_ s - x) dsdt, \qquad x \in R^ d,\;\varepsilon > 0,$ by series of multiple Wiener-Itô integrals for any dimension $$d$$, and from this they deduce that $$\alpha(x) = \lim_{\varepsilon \downarrow 0} \alpha_ \varepsilon (x)$$ can be considered as a well-defined object in the Sobolev space $$D^{2,\gamma}$$ over the canonical Wiener space of any order $$\gamma < (4-d)/2$$ if $$x \neq 0$$. From here they rederive the existence of the double intersection local time for $$d = 2,3$$, shown by J. Rosen [Ann. Probab. 14, 1245-1251 (1986; Zbl 0617.60079), Stochastic Processes Appl. 23, 131-141 (1986; Zbl 0612.60070), and in: Séminaire de probabilités XX, Lect. Notes Math. 1204, 515-531 (1986; Zbl 0611.60065)]; for $$d\geq 4$$ the results are new and could be compared with the smoothed versions considered by M. Yor [in: Séminaire de probabilités XIX, Lect. Notes Math. 1123, 332-349 (1985; Zbl 0563.60073)].
For $$x \to 0$$, $$\alpha(x)$$ does not converge in any Sobolev space, $$\alpha$$ has to be renormalised by a suitable deterministic function in order to get the convergence. The difficulty consists in a control of the norms of the projections on the eigenspaces of the Ornstein-Uhlenbeck operator. In case $$d = 2$$, S. R. S.Varadhan’s renormalization (1969) is improved, in case $$d = 3$$, versions of M. Yor’s renormalization (loc. cit.) turn up, and also the case $$d \geq 4$$ is dealt with.
Reviewer: R.Buckdahn (Brest)

### MSC:

 60H05 Stochastic integrals 60H07 Stochastic calculus of variations and the Malliavin calculus 60J55 Local time and additive functionals

### Citations:

Zbl 0617.60079; Zbl 0612.60070; Zbl 0611.60065; Zbl 0563.60073
Full Text:

### References:

 [1] Bass, R.; Koshnevisan, D., Intersection local times and Tanaka formulas (1992), Univ. of Washington, Preprint [2] Bolthausen, E., On the construction of the three dimensional polymer measure (1990), Univ. Zürich, Preprint [3] Bouleau, N.; Hirsch, F., Dirichlet Forms and Analysis on Wiener Space (1991), W. de Gruyter: W. de Gruyter Berlin · Zbl 0748.60046 [4] Dvoretzky, A.; Erdös, P.; Kakutani, S., Double points of paths of Brownian motion in the plane, Bull. Res. Council Israel Sect., F3, 364-371 (1954) [5] Dvoretzky, A.; Erdös, P.; Kakutani, S.; Taylor, S. J., Triple points of the Brownian motion in 3-space, (Proc. Cambridge Philos. Soc., 53 (1957)), 856-862 · Zbl 0208.44103 [6] Dynkin, E. B., Polynomials of the occupation field and related random fields, J. Funct. Anal., 58, 20-52 (1984) · Zbl 0552.60075 [7] Dynkin, E. B., Regularized self-intersection local times of planar Brownian motion, Ann. Probab., 16, 58-74 (1988) · Zbl 0641.60085 [8] Dynkin, E. B., Self-intersection gauge for random walks and for Brownian motion, Ann. Probab., 16, 1-57 (1988) · Zbl 0638.60081 [9] Geman, D.; Horowitz, J.; Rosen, J., A local time analysis of intersections of Brownian paths in the plane, Ann. Probab., 12, 86-107 (1984) · Zbl 0536.60046 [10] He, S. W.; Yang, W. Q.; Yao, R. Q.; Wang, J. G., Local times of self-intersection for multidimensional Brownian motion (1993), Preprint [11] Hida, T.; Potthoff, J.; Streit, L., White Noise (1993), Kluwer · Zbl 0722.60034 [13] Imkeller, P.; Weisz, F., The asymptotic behaviour of local times and occupation integrals of the $$N$$-parameter Wiener process in $$R^d$$, Probab. Theory Relat. Fields, 98, 47-75 (1994) · Zbl 0794.60046 [14] Imkeller, P.; Weisz, F., Critical dimensions for the existence of self intersection local times of the Brownian sheet in $$R^d (1993)$$, Preprint [15] Kuo, H. H., Donsker’s delta function as a generalized Brownian functional and its application, (Lecture Notes in Control and Information Sciences, Vol. 49 (1983), Springer: Springer Berlin), 167-178 [16] Le Gall, J. F., Sur le temps local d’intersection du mouvement brownien plan et la méthode de renormalisation de Varadhan, (Sém. de Prob. XIX. Sém. de Prob. XIX, 1983/84, LNM 1123 (1985), Springer: Springer Berlin), 314-331 · Zbl 0563.60072 [17] Le Gall, J. F., Sur la saucisse de Wiener et les points multiples du mouvement brownien, Ann. Probab., 14, 1219-1244 (1986) · Zbl 0621.60083 [18] Lévy, P., Le mouvement brownien plan, Amer. J. Math., 62, 440-487 (1940) · JFM 66.0619.02 [19] Lyons, T. J., The critical dimension at which quasi-every Brownian motion is self-avoiding, Adv. in Appl. Probab., 87-99 (1986), (Spec. Suppl. 1986) · Zbl 0609.60087 [20] Nualart, D.; Vives, J., Smoothness of Brownian local times and related functionals, Potential Analysis, 1, 257-263 (1992) · Zbl 0776.60092 [22] Penrose, M. D., On the existence of self-intersections for quasi-every Brownian path in space, Ann. Probab., 17, 482-502 (1989) · Zbl 0714.60067 [23] Rosen, J., A local time approach to the self-intersections of Brownian paths in space, Comm. Math. Phys., 88, 327-338 (1983) · Zbl 0534.60070 [24] Rosen, J., Tanaka’s formula and renormalization for intersections of planar Brownian motion, Ann. Probab., 14, 1245-1251 (1986) · Zbl 0617.60079 [25] Rosen, J., A renormalized local time for multiple intersections of planar Brownian motion, (Sém. de Prob. XX. Sém. de Prob. XX, 1984/85, LNM 1204 (1986), Springer: Springer Berlin), 515-531 [26] Shieh, N. R., White noise analysis and Tanaka formula for intersections of planar Brownian motion, Nagoya Math. J., 122, 1-17 (1991) · Zbl 0759.60041 [27] Szegö, G., Orthogonal polynomials, (Amer. Math. Soc. Colloquium Publications, Vol. XXIII (1939), Amer. Math. Soc: Amer. Math. Soc New York) · JFM 61.0386.03 [28] Szymanzik, K., Euclidean quantum field theory, (Jost, R., Local Quantum Theory (1969), Academic: Academic New York) [29] Varadhan, S. R.S., Appendix to “Euclidean quantum field theory” by K. Szymanzik, (Jost, R., Local Quantum Theory (1969), Academic: Academic New York) [30] Watanabe, H., The local time of self-intersections of Brownian motions as generalized Brownian functionals, Letters in Math. Physics, 23, 1-9 (1991) · Zbl 0748.60068 [31] Watanabe, H., Donsker’s delta function and its applications in the theory of white noise analysis, (Stochastic Processes, A Festschrift in honour of Gopinath Kallianpur (1993), Springer: Springer Berlin) · Zbl 0791.60030 [32] Watanabe, S., Lectures on Stochastic Differential Equations and Malliavin Calculus (1984), Springer: Springer Berlin · Zbl 0546.60054 [33] Westwater, J., On Edwards’ model for long polymer chains, Comm. Math. Phys., 72, 131-174 (1980) · Zbl 0431.60100 [34] Wolpert, R., Wiener path intersection and local time, J. Funct. Anal., 30, 329-340 (1978) · Zbl 0403.60069 [35] Yor, M., Compléments aux formules de Tanaka-Rosen, (Sém. de Prob. XIX. Sém. de Prob. XIX, 1983/84. LNM 1123 (1985), Springer: Springer Berlin), 332-348 · Zbl 0563.60073 [36] Yor, M., Renormalisation et convergence en loi pour les temps locaux d’intersection du mouvement brownien dans $$R^3$$, (Sém. de Prob. XIX. Sém. de Prob. XIX, 1983/84. LNM 1123 (1985), Springer: Springer Berlin), 350-365 · Zbl 0569.60075
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