Polynomials of the occupation field and related random fields (Q801611)

This article is a direct continuation of the author's paper ibid. 55, 344-376 (1984; Zbl 0533.60061). For a good symmetric transition density \((p_ t)\) on a measure space E two random (but maybe generalized) fields are associated. They are free (Gaussian) field \(\Phi\) and the occupation field T, the latter characterizing the total time spent at x. If p is the Brownian motion on \({\mathbb{R}}^ d\) with \(d\leq 2\) and with killing rate \(k>0\), then the ''renormalized'' Wick's n-th power of T can be defined. This generalized random \(field\vdots T^ n_ z\vdots\) describes the size of the random set \(\{(u_ 1...u_ n):w(u_ 1)=w(u_ 2)=...`w(u_ n)=z\}\) and hence characterizes the space location of self- intersections of order n. Actually the product of \(\vdots T^ j_ z\vdots\) and \(\vdots \xi^ i_ f\vdots\) is defined by a suitable limiting procedure (Theorem 1.3 in this article), and the main results obtained are valid for every good \((p_ t)\) (specified in {\S} 1.8).