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

Bally, Vlad; Stoica, Lucretiu

Ann. Probab. 15, 241-262 (1987).

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

Keywords:

local times; additive functional; occupation time; downcrossing approximating models