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

Peter Imkeller and Ferenc Weisz
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Peter Imkeller: Humboldt Universität zu Berlin
Ferenc Weisz: Eötvös L. University

Journal of Theoretical Probability, 1999, vol. 12, issue 3, 721-737

Abstract: 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.

Keywords: N-parameter Brownian motion; self-intersection local time; multiple stochastic integrals; canonical Dirichlet structure (search for similar items in EconPapers)
Date: 1999
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DOI: 10.1023/A:1021627815734

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