×

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

Citations:

Zbl 0467.60069
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Dynkin, E. B., (Markov Processes, Vols. 1 and 2 (1965), Springer-Verlag: Springer-Verlag Berlin, Göttingen, Heidelberg) · Zbl 0132.37901
[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, (Çinlar, E.; Chung, K. L.; Getoor, R. K., Seminar on Stochastic Processes. Seminar on Stochastic Processes, 1983 (1984), Birkhäuser: Birkhäuser Boston/Basel/Stuttgart) · 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, (Azéma, J.; Yor, M., Séminaire de Probabilités 19. Séminaire de Probabilités 19, 1983/84 (1985), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York/Tokyo), 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.; J. Rosen, A representation for the intersection local time of Brownian motion in space, Ann. Probab. · 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, (Jost, R., Local Quantum Theory (1969), Academic Press: Academic Press New York/London), 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, (Azéma, J.; Yor, M., Séminaire de Probabilités 19. Séminaire de Probabilités 19, 1983/84 (1985), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York/Tokyo), 332-349 · Zbl 0563.60073
[21] Yor, M., Renormalisation et convergence en loi pour les temps locaux d’intersection du mouvement Brownien dans \(\textbf{R}^3\), (Azéma, J.; Yor, M., Séminaire de Probabilités 19. Séminaire de Probabilités 19, 1983/84 (1985), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York/Tokyo), 350-365 · Zbl 0569.60075
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.