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Polynomials of the occupation field and related random fields. (English) Zbl 0552.60075
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 § 1.8).
Reviewer: W.Yang

60J65 Brownian motion
60J55 Local time and additive functionals
60G20 Generalized stochastic processes
60H05 Stochastic integrals
Full Text: DOI
[1] Dynkin, E.B, Gaussian and non-Gaussian random fields associated with Markov processes, J. funct. anal., 55, (1984) · Zbl 0533.60061
[2] Meyer, P.-A, Limites médiates d’après mokobodzki, () · Zbl 0262.28005
[3] Watson, G.N, A treatise on the theory of Bessel functions, (1952), Cambridge Univ. Press Cambridge · Zbl 0849.33001
[4] Yosida, K, Functional analysis, (1978), Springer-Verlag Berlin/Heidelberg/New York · Zbl 0152.32102
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