Generalized random fields related to self-intersections of the Brownian motion - Dynkin, E. B. (1986)
Let T(k)(ε)(λ; t(1),..., t(k)) = ρ(X(t)(1))q(ε)(X(t)(2) - X(t)(1))... q(ε)(X(t(k)) - X(t(k))-1), where X(t) is a Brownian motion in R(2), λ(dx) = ρ(x)dx, and q(ε) converges to Dirac's δ function as ε ↓ 0. The self-intersection local times of order k are described by a generalized random field T(k)(λ; t(1),..., t(k)) = lim(ε↓0)T(k)(ε)(λ; t(1),..., t(k)) for 0 < t(1) <... < t(k). The field “blows up” as t(i) - t(j) → 0 for some i ≠ j. I show that with a proper choice of the coefficients B(k)(l)(ε), a generalized random field [unk] (k)(λ; t(1),..., t(k)) = lim(ε↓0) [T(k)(ε)(λ; t(1),..., t(k)) + Σ(l=1)(k-1) [B(k)(l)(ε)T(l)(ε)](λ; t(1),..., t(k))] is well defined for all 0 ≤ t(1) ≤... ≤ t(k) and it coincides with T(k)(λ; t(1),..., t(k)) for t(1) <... < t(k).