## A class of Markov processes which admit local times.(English)Zbl 0615.60069

A class of standard processes which admit local times at each point is considered. The following regularity properties are assumed: $$T_ x\to T_ a=0$$ (as $$x\to a)$$ in $$P^ a$$-probability and $$P^ a(T_ b<\infty)>0$$ for all pairs of points a,b $$(T_ x=\inf \{t>0:$$ $$X_ t=x\})$$. The class under consideration turns out to be very large. It is already known that a wide class of processes with independent increments fulfill our hypothesis. We also observe that the class is left invariant by the usual transformations: time change, subprocess and u-process (h- path) transformations.
The first important result of the paper is that every continuous additive functional may be represented as a mixture (integral) of local times. This theorem is used to prove two further results. The first one asserts that every process in the class has a dual process which remains in the class. Particularly Hunt’s hypothesis (F) is satisfied. The second one generalises the occupation time and downcrossing approximating models. Such approximation theorems are proved for a C.A.F. whose representing measure is given.

### MSC:

 60J55 Local time and additive functionals 60G50 Sums of independent random variables; random walks
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