Fictitious states,coupled laws and local time

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 ^y of 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 ^y in terms of g ^y may 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.