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Critical Dimensions for the Existence of Self-Intersection Local Times of the N-Parameter Brownian Motion in R d

Author

Listed:
  • Peter Imkeller

    (Humboldt Universität zu Berlin)

  • Ferenc Weisz

    (Eötvös L. University)

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 $$(W_t )_{t \in } [0,1]^N $$ 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.

Suggested Citation

  • Peter Imkeller & Ferenc Weisz, 1999. "Critical Dimensions for the Existence of Self-Intersection Local Times of the N-Parameter Brownian Motion in R d," Journal of Theoretical Probability, Springer, vol. 12(3), pages 721-737, July.
    • Handle: RePEc:spr:jotpro:v:12:y:1999:i:3:d:10.1023_a:1021627815734
    DOI: 10.1023/A:1021627815734
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