In a paper of the 5th Symposium of Berkeley, Motoo has proposed the following problem : Let $G$ be a dense open set in a compact $S$. Then describe all processes such that their behaviours before reacting $3G$, the boundary of $G$, is the same as that of a given minimal process.

Much progress of the solving of one aspect of the problem -actually in a generalized form- is due to Getoor and Sharpe [21], using Chung's and Azema's sweeping out techniques and finally obtaining "last exit decomposition" and contion- ning formulas. Further work has been done by Meyer [17] . In our paper, we start again, for more clearness, with Meyer's presentation, but we are mostly interested on the description of processes by means of compactification. More general theorems than Getoor and Sharpe are obtained, and simple interpretations are given to some concepts.

A chapter is reserved to the $R^{n}$ case.