Penalising symmetric stable L\'evy paths

Limit theorems for the normalized laws with respect to two kinds of weight functionals are studied for any symmetric stable L\'evy 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 L\'evy counterpart of the universal $\sigma$-finite measure, found in [9] and [10], which unifies the corresponding limit theorems in the Brownian setup for which $\alpha =2$.