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 $\beta\le\sigma<\displaystyle\min \Big\{ { 3\over 2}\beta, d\Big\}$, 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.
Categories: math.PR