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- ItemSelf-intersection local times of random walks : exponential moments in subcritical dimensions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Becker, Mathias; König, WolfgangFix 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.
- ItemUpper tails of self-intersection local times of random walks : survey of proof techniques(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) König, WolfgangWe discuss the logarithmic asymptotics for the upper tails of self-intersection local times of random walks on Zd. This topic has been studied a lot in the last decade, since it is a natural question, and a rich phenemonology of critical behaviours of the random walk arises, depending on the dimension, the intersection parameter, the scale, and the type of the random process. Furthermore, the question is technically difficult to handle, due to bad continuity and boundedness properties of the self-intersection local time. A couple of different techniques for studying self-intersections have been introduced yet, wich turned out to be more or less fruitful in various situations. It is the goal of this note to survey and compare some of the most fruitful techniques used in recent years