Summary
The work of Ray and Neveu has established that, for any transition function P on a countable set E, (i) there exists a best possible entrance boundary E + supporting a right continuous, strong Markov process X with transition function P and that (ii) the points y of E + are in one-one correspondence with the extremal entrance laws g yof P. Here, it is shown that, if a point y of E + is regular for itself, then the derived characteristic f y of the local time at y is a regular extremal entrance law “coupled” with g y in the sense of Neveu. Further, coupled laws arise only in this fashion. By using excursion theory, a simple explicit formula for f yin terms of g ymay be obtained. The paper contains a conjecture about the intrinsic character of the Ray-Neveu topology and an example which shows emphatically that, in general, local time is not a derivative of occupation time.
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Williams, D. Fictitious states, coupled laws and local time. Z. Wahrscheinlichkeitstheorie verw Gebiete 11, 288–310 (1969). https://doi.org/10.1007/BF00531652
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DOI: https://doi.org/10.1007/BF00531652