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

Abstract：

: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

journal_name

Proc Natl Acad Sci U S A

journal_title

Proceedings of the National Academy of Sciences of the United States of America

authors

Dynkin EB

doi

10.1073/pnas.83.11.3575

subject

Has Abstract

pub_date

1986-06-01 00:00:00

pages

3575-6

issue

11

eissn

0027-8424

issn

1091-6490

journal_volume

83

pub_type

杂志文章

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