Large deviations and renormalization for Riesz potentials of stable intersection measures

Abstract

We study the object formally defined as γ\big([0,t]^{2}\big)=\int\int_{[0,t]^{2}} | X_{s}- X_{r}|^{-σ} dr ds-E\int\int_{[0,t]^{2}} | X_{s}- X_{r}|^{-σ} dr ds, where $X_t$ is the symmetric stable processes of index $0<\beta \le 2$ in $R^d$. When $\beta \le \sigma <\min \{\frac{3}{2}\beta ,d\}$, 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.

References in corpus (1)

• Large deviations for renormalized self-intersection local times of stable processes