Local Time, Excursions, and Additive Functionals
Tanaka’s formula and semimartingale local time; occupation density, continuity and approximation; regenerative sets and processes; excursion local time and Poisson process; Ray-Knight theorem; excessive functions and additive functionals; local time at a regular point; additive functionals of Brownian motion
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- Local time of Brownian motion at a fixed point was discovered and explored by Levy (1939), who devised several explicit constructions, mostly of the type of Proposition 22.12. Much of Lévy’s analysis is based on the observation in Corollary 22.3. The elementary Lemma 22.2 is due to Skorohod (1961–62). Formula (1), first noted for Brownian motion by Tanaka (1963), was taken by Meyer (1976) as the basis for a general semi-martingale approach. The general Itô-Tanaka formula in Theorem 22.5 was obtained independently by Meyer (1976) and Wang (1977). Trotter (1958b) proved that Brownian local time has a jointly continuous version, and the extension to general continuous semimartingales in Theorem 22.4 was obtained by Yor (1978).Google Scholar
- Modern excursion theory originated with the seminal paper of Ito (1972), which was partly inspired by earlier work of Lévy (1939). In particular, Ito proved a version of Theorem 22.11, assuming the existence of local time. Horowitz (1972) independently studied regenerative sets and noted their connection with subordinators, equivalent to the existence of a local time. A systematic theory of regenerative processes was developed by Maisonneuve (1974). The remarkable Theorem 22.17 was discovered independently by Ray (1963) and Knight (1963), and the present proof is essentially due to Walsh (1978). Our construction of the excursion process is close in spirit to Lévy’s original ideas and to those in Greenwood and Pitman (1980).Google Scholar
- Elementary additive functionals of integral type had been discussed extensively in the literature when Dynkin proposed a study of the general case. The existence Theorem 22.23 was obtained by Volkonsky (1960), and the construction of local time in Theorem 22.24 dates back to Blumen-thal and Getoor (1964). The integral representation of CAFs in Theorem 22.25 was proved independently by Volkonsky (1958, 1960) and McK-ean and Tanaka (1961). The characterization of additive functionals in terms of suitable measures on the state space dates back to Meyer (1962), and the explicit representation of the associated measures was found by Revuz (1970) after special cases had been considered by Hunt (1957–58).Google Scholar
- An excellent introduction to local time appears in Karatzas and Shreve (1991). The books by Itô and McKean (1965) and Revuz and Yor (1999) contain an abundance of further information on the subject. The latter text may also serve as a good introduction to additive func-tionals and excursion theory. For more information on the latter topics, the reader may consult Blumenthal and Getoor (1968), Blumenthal (1992), and Dellacherie et al. (1992).Google Scholar