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Optimal Bounds for the Variance of Self-Intersection Local Times

George Deligiannidis & Sergey Utev

International Journal of Stochastic Analysis2016https://doi.org/10.1155/2016/5370627article
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0.26

Abstract

For a Zd-valued random walk (Sn)n∈N0, let l(n,x) be its local time at the site x∈Zd. For α∈N, define the α-fold self-intersection local time as Ln(α)≔∑xl(n,x)α. Also let LnSRW(α) be the corresponding quantities for the simple random walk in Zd. Without imposing any moment conditions, we show that the variance of the self-intersection local time of any genuinely d-dimensional random walk is bounded above by the corresponding quantity for the simple symmetric random walk; that is, var(Ln(α))=O(var⁡(LnSRW(α))). In particular, for any genuinely d-dimensional random walk, with d≥4, we have var⁡(Ln(α))=O(n). On the other hand, in dimensions d≤3 we show that if the behaviour resembles that of simple random walk, in the sense that lim infn→∞var⁡Lnα/var⁡(LnSRW(α))>0, then the increments of the random walk must have zero mean and finite second moment.

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https://doi.org/https://doi.org/10.1155/2016/5370627

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@article{george2016,
  title        = {{Optimal Bounds for the Variance of Self-Intersection Local Times}},
  author       = {George Deligiannidis & Sergey Utev},
  journal      = {International Journal of Stochastic Analysis},
  year         = {2016},
  doi          = {https://doi.org/https://doi.org/10.1155/2016/5370627},
}

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