The inverse of a continuous additive functional

Let X be a standard process and A be a continuous additive functional of X.The inverse of A is defined by τ t -mί{sΆ s > t}.The aim of this paper is to prove that the process τ has conditionally independent increments with respect to the σ -algebra generated by the time changed process X t = X Tt .However these increments are not necessarily stationary.Another interesting result is derived: the continous part of the process T is a continuous additive functional of the process X.The existence of regular conditional probabilities permits to consider the process r as an additive process and under a necessary and sufficient condition, it is in fact a Levy process with increasing paths.The general theory of such processes is then used to obtain a Levy representation of the jumps of the process r.