### Abstract

Fix two rectangles A, B in [0, 1]^{N}. Then the size of the random set of double points of the N-parameter Brownian motion (W_{t})_{t∈[0, 1]N} in R^{d}, i.e, the set of pairs (s, t), where s ∈ A, t ∈ B, and W_{s} = W_{t}, can be measured as usual by a self-intersection local time. If A = B, we show that the critical dimension below which self-intersection local time does not explode, is given by d = 2N. If A ∩ B is a p-dimensional rectangle, it is 4N - 2p (0 ≤ p ≤ N). If A ∩B = ∅, it is infinite. In all cases, we derive the rate of explosion of canonical approximations of self-intersection local time for dimensions above the critical one, and determine its smoothness in terms of the canonical Dirichlet structure on Wiener space.

Original language | English |
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Pages (from-to) | 721-737 |

Number of pages | 17 |

Journal | Journal of Theoretical Probability |

Volume | 12 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jan 1 1999 |

### Keywords

- Canonical Dirichlet structure
- Multiple stochastic integrals
- N-parameter Brownian motion
- Self-intersection local time

### ASJC Scopus subject areas

- Statistics and Probability
- Mathematics(all)
- Statistics, Probability and Uncertainty