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Marginal densities of the “true” self-repelling motion

Author

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  • Dumaz, Laure
  • Tóth, Bálint

Abstract

Let X(t) be the true self-repelling motion (TSRM) constructed by Tóth and Werner (1998) [22], L(t,x) its occupation time density (local time) and H(t):=L(t,X(t)) the height of the local time profile at the actual position of the motion. The joint distribution of (X(t),H(t)) was identified by Tóth (1995) [20] in somewhat implicit terms. Now we give explicit formulas for the densities of the marginal distributions of X(t) and H(t). The distribution of X(t) has a particularly surprising shape: its density has a sharp local minimum with discontinuous derivative at 0. As a consequence we also obtain a precise version of the large deviation estimate of Dumaz (2011) [5].

Suggested Citation

  • Dumaz, Laure & Tóth, Bálint, 2013. "Marginal densities of the “true” self-repelling motion," Stochastic Processes and their Applications, Elsevier, vol. 123(4), pages 1454-1471.
    • Handle: RePEc:eee:spapps:v:123:y:2013:i:4:p:1454-1471
    DOI: 10.1016/j.spa.2012.11.011
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