Generalized random fields related to self-intersections of the Brownian motion

AUTOR(ES)

Dynkin, E. B.

RESUMO

Let Tkε(λ; t1,..., tk) = ρ(Xt1)qε(Xt2 - Xt1)... qε(Xtk - Xtk-1), where Xt is a Brownian motion in R2, λ(dx) = ρ(x)dx, and qε converges to Dirac's δ function as ε ↓ 0. The self-intersection local times of order k are described by a generalized random field Tk(λ; t1,..., tk) = limε↓0Tkε(λ; t1,..., tk) for 0 < t1 <... < tk. The field “blows up” as ti - tj → 0 for some i ≠ j. I show that with a proper choice of the coefficients Bkl(ε), a generalized random field [unk] k(λ; t1,..., tk) = limε↓0 [Tkε(λ; t1,..., tk) + Σl=1k-1 [Bkl(ε)Tlε](λ; t1,..., tk)] is well defined for all 0 ≤ t1 ≤... ≤ tk and it coincides with Tk(λ; t1,..., tk) for t1 <... < tk.

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