A Class of Markov Processes which Admit Local Times

Abstract

A class of standard processes which admit local times at each point is considered. The following regularity properties are assumed: $T_x \rightarrow T_a = 0$ (as $x \rightarrow a$) 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.