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Multiple path integrals. (English) Zbl 0604.60075
The notion of additive functional of order k for a Markov process is introduced and studied in a general setting. Usual additive functionals are recovered in the case $$k=1$$. An important special case of these generalized additive functionals is the self-intersection local times of Brownian motion, which have been studied in many recent papers. A relationship is described between additive functionals of order k for one process, and additive functionals of k independent processes, which have been studied by the author in a previous paper [J. Funct. Anal. 42, 64- 101 (1981; Zbl 0467.60069)].
Reviewer: J.F.Le Gall

##### MSC:
 60J55 Local time and additive functionals 60J25 Continuous-time Markov processes on general state spaces 60J60 Diffusion processes
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##### References:
 [1] Dynkin, E.B, () [2] Dynkin, E.B, Markov systems and their additive functionals, Ann. probab., 5, 653-677, (1977) · Zbl 0379.60076 [3] Dynkin, E.B, Additive functionals of several time-reversible Markov processes, J. fund. anal., 42, 1, 64-101, (1981) · Zbl 0467.60069 [4] Dynkin, E.B, Green’s and Dirichlet spaces associated with fine Markov processes, J. funct. anal., 47, 3, 381-418, (1982) · Zbl 0488.60083 [5] Dynkin, E.B, Local times and quantum fields, () · Zbl 0554.60058 [6] Dynkin, E.B, Polynomials of the occupation field and related random fields, J. funct. anal., 58, 1, 20-52, (1984) · Zbl 0552.60075 [7] Dynkin, E.B, Random fields associated with multiple points of the Brownian motion, J. funct. anal., 62, 3, (1985) · Zbl 0579.60081 [8] Dynkin, E.B; Getoor, R.K, Additive functionals and entrance laws, J. funct. anal., 62, 2, 221-265, (1985) · Zbl 0574.60082 [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] Getoor, R.K; Sharpe, M.J, Naturality, standardness, and weak duality for Markov processes, Z. wahrsch. verw. gebiete, 67, 1-62, (1984) · Zbl 0553.60070 [11] Le Gall, J.F, Sur la sucisse de Wiener et LES points multiples du mouvement brownien, (1984), [Preprint] [12] Le Gall, J.F, Sur le temps local d’intersection du mouvement brownien plan et la méthode de renormalisation de varadhan, (), 314-331 · Zbl 0563.60072 [13] 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 [14] Rosen, J, Tanaka’s formula and renormalization for intersections of planar Brownian motion, (1984), [Preprint] [15] J. Rosen, A representation for the intersection local time of Brownian motion in space, Ann. Probab., to appear. · Zbl 0561.60086 [16] Rosen, J, Joint continuity of the intersection local times of Markov processes, (1985), [Preprint] [17] Varadhan, S.R.S, Appendix to Euclidean quantum field theory, (), by K. Symanzik · Zbl 0980.60002 [18] Westwater, M.J, On edwards’ model for long polymer chains, Comm. math. phys., 72, 131-174, (1980) · Zbl 0431.60100 [19] Wolpert, R.L, Wiener path intersections and local time, J. fund. anal., 30, 329-340, (1978) · Zbl 0403.60069 [20] Yor, M, Compléments aux formules de Tanaka-Rosen, (), 332-349 · Zbl 0563.60073 [21] Yor, M, Renormalisation et convergence en loi pour LES temps locaux d’intersection du mouvement brownien dans R3, (), 350-365 · Zbl 0569.60075
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