TY - JOUR

T1 - Critical Dimensions for the Existence of Self-Intersection Local Times of the N-Parameter Brownian Motion in Rd

AU - Imkeller, Peter

AU - Weisz, Ferenc

PY - 1999/1/1

Y1 - 1999/1/1

N2 - 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 (Wt)t∈[0, 1]N in Rd, i.e, the set of pairs (s, t), where s ∈ A, t ∈ B, and Ws = Wt, 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.

AB - 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 (Wt)t∈[0, 1]N in Rd, i.e, the set of pairs (s, t), where s ∈ A, t ∈ B, and Ws = Wt, 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.

KW - Canonical Dirichlet structure

KW - Multiple stochastic integrals

KW - N-parameter Brownian motion

KW - Self-intersection local time

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U2 - 10.1023/A:1021627815734

DO - 10.1023/A:1021627815734

M3 - Article

AN - SCOPUS:0033471173

VL - 12

SP - 721

EP - 737

JO - Journal of Theoretical Probability

JF - Journal of Theoretical Probability

SN - 0894-9840

IS - 3

ER -