WIAS Preprint No. 1532, (2010)
Self-intersection local times of random walks: Exponential moments in subcritical dimensions
- Becker, Mathias
- König, Wolfgang
2010 Mathematics Subject Classification
- Self-intersection local time, upper tail, Donsker-Varadhan large deviations, variational formula, Gagliardo-Nirenberg inequality
Fix $p>1$, not necessarily integer, with $p(d-2)0$ that are bounded from above, possibly tending to zero. The speed is identified in terms of mixed powers of $t$ and $theta_t$, and the precise rate is characterized in terms of a variational formula, which is in close connection to the it Gagliardo-Nirenberg inequality. As a corollary, we obtain a large-deviation principle for $ ell_t _p/(t r_t)$ for deviation functions $r_t$ satisfying $t r_tggE[ ell_t _p]$. Informally, it turns out that the random walk homogeneously squeezes in a $t$-dependent box with diameter of order $ll t^1/d$ to produce the required amount of self-intersections. Our main tool is an upper bound for the joint density of the local times of the walk.
- Probab. Theory Related Fields, 154 (2012) pp. 585--605.