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Random fields associated with multiple points of the Brownian motion. (English) Zbl 0579.60081
Let \(\Gamma =\{(t_ 1,...,t_ n,z): X_{t_ 1}=...=X_{t_ n}=z\}\) be the random set of self-intersections for the Brownian motion \(X_ t\) in \({\mathbb{R}}^ d\), \(\Gamma =\emptyset\) a.s. except some cases. In each of these cases, a family of random measures \(M_{\lambda}\) concentrated on \(\Gamma\) is constructed (\(\lambda\) takes values in a certain class of measures on \({\mathbb{R}}^ d)\). Measures \(M_{\lambda}\) characterise the time-space location of the set \(\Gamma\). These measures \(M_{\lambda}\) ”explode” on the hyper-planes \(t_ i=t_ j\), so the main part of the paper is devoted to the problem of regularization. In the case \(n=2\) the answer is given by the theorem I.4, for \(n>2\) the answer is not known.
Reviewer: A.N.Radchenko

MSC:
60J65 Brownian motion
60G60 Random fields
60J55 Local time and additive functionals
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