Rogers, L. C. G.; Walsh, J. B. The intrinsic local time sheet of Brownian motion. (English) Zbl 0722.60079 Probab. Theory Relat. Fields 88, No. 3, 363-379 (1991). McGill showed that the intrinsic local time process \(\tilde L(t,x)\), \(t\geq 0\), \(x\in {\mathbb{R}}\), of one-dimensional Brownian motion is, for fixed \(t>0\), a supermartingale in the space variable, and derived an expression for its Doob-Meyer decomposition. This expression referred to the derivative of some process which was not obviously differentiable. In this paper, we provide an independent proof of the result, by analysing the local time of Brownian motion on a family of decreasing curves. The ideas involved are best understood in terms of stochastic area integrals with respect to the Brownian local time sheet, and we develop this approach in a companion paper. However, the result mentioned above admits a direct proof, which we give here; one is inevitably drawn to look at the local time process of a Dirichlet process which is not a semimartingale. Reviewer: L.C.G.Rogers Cited in 6 Documents MSC: 60J55 Local time and additive functionals 60J65 Brownian motion Keywords:local time process; Doob-Meyer decomposition; Brownian local time sheet; Dirichlet process; semimartingale × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Bertoin, J., Temps locaux et intégration stochastique pour les processus de Dirichlet. Séminaire de Probabilités XXI, 191-205 (1987), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York · Zbl 0616.60073 [2] Dellacherie, C.; Meyer, P. A., Probabilités et Potentiel, Chap. I-IV (1975), Paris: Hermann, Paris · Zbl 0323.60039 [3] Jeulin, T.; Çinlar, E.; Chung, K. L.; Getoor, R. K., Ray-Knight’s theorem on Brownian local times and Tanaka’s formula (1984), Boston: Birkhäuser, Boston · Zbl 0561.60077 [4] McGill, P., Calculation of some conditional excursion formulae, Z. Wahrscheinlich-keitstheor. Verw. Geb., 61, 255-260 (1982) · Zbl 0495.60081 [5] McGill, P., Integral representation of martingales in the Brownian excursion filtration, 465-502 (1986), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York · Zbl 0635.60057 [6] Meyer, P. A., Un cours sur les intégrales stochastiques. Séminaire de Probabilités X, 245-400 (1976), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York · Zbl 0374.60070 [7] [RWa] Rogers, L.C.G., Walsh, J.B.: Local time and stochastic area integrals. Ann. Probab. (to appear) · Zbl 0729.60073 [8] Rogers, L. C.G.; Walsh, J. B.; Fitzsimmons, P.; Williams, R. J., A (t, Bt) is not a semimartingale (1991), Boston: Birkhäuser, Boston · Zbl 0721.60089 [9] Rogers, L. C.G.; Williams, D., Diffusions, Markov processes, and martingales, vol. 2 (1987), Chichester: Wiley, Chichester · Zbl 0627.60001 [10] Walsh, J. B., Excursions and local time. Temps locaux. Astérisque52-53 (1978), Paris: Société Mathématique de France, Paris [11] Walsh, J. B.; Çinlar, E.; Chung, K. L.; Getoor, R. K., Stochastic integration with respect to local time (1984), Boston: Birkhäuser, Boston [12] Williams, D., Conditional excursion theory, 490-494 (1979), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York · Zbl 0422.60058 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.