Penalising symmetric stable Lévy paths

Limit theorems for the normalized laws with respect to two kinds of weight functionals are studied for any symmetric stable Lévy process of index 1 < \alpha \le 2 . The first kind is a function of the local time at the origin, and the second kind is the exponential of an occupation time integral. Special emphasis is put on the role played by a stable Lévy counterpart of the universal \sigma -finite measure, found in [9] and [10], which unifies the corresponding limit theorems in the Brownian setup for which \alpha = 2 .