Large deviations and renormalization for Riesz potentials of stable intersection measures

We study the object formally defined as \gamma\big([0,t]^{2}\big)=\int\int_{[0,t]^{2}} | X_{s}- X_{r}|^{-\sigma} dr ds-E\int\int_{[0,t]^{2}} | X_{s}- X_{r}|^{-\sigma} dr ds, where $X_t$ is the symmetric stable processes of index $0<\beta \le 2$ in $R^d$. When $\beta \le \sigma <{\displaystyle \min \{\frac{3}{2}\beta ,d\}}$, this has to be defined as a limit, in the spirit of renormalized self-intersection local time. We obtain results about the large deviations and laws of the iterated logarithm for $\gamma$. This is applied to obtain results about stable processes in random potentials.