Generalized random fields related to self-intersections of the Brownian motion.
Let T(k) (epsilon)(lambda; t(1),..., t(k)) = rho(X(t) (1))q(epsilon)(X(t) (2) - X(t) (1))... q(epsilon)(X(t(k) ) - X(t(k) )-1), where X(t) is a Brownian motion in R(2), lambda(dx) = rho(x)dx, and q(epsilon) converges to Dirac's delta function as epsilon downward arrow 0. The self-intersection local times of order k are described by a generalized random field T(k)(lambda; t(1),..., t(k)) = lim(epsilon downward arrow0)T(k) (epsilon)(lambda; t(1),..., t(k)) for 0 < t(1) <... < t(k). The field "blows up" as t(i) - t(j) --> 0 for some i not equal j. I show that with a proper choice of the coefficients B(k) (l)(epsilon), a generalized random field [unk] (k)(lambda; t(1),..., t(k)) = lim(epsilon downward arrow0) [T(k) (epsilon)(lambda; t(1),..., t(k)) + Sigma(l=1) (k-1) [B(k) (l)(epsilon)T(l) (epsilon)](lambda; t(1),..., t(k))] is well defined for all 0 </= t(1) </=... </= t(k) and it coincides with T(k)(lambda; t(1),..., t(k)) for t(1) <... < t(k).
Dynkin
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EB
Department of Mathematics, Cornell University, Ithaca, NY 14853.