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Chaos expansions of double intersection local time of Brownian motion in \(\mathbb{R}^ d\) and renormalization. (English) Zbl 0822.60048

Let \(W = \{W_ t,\;t \in [0,1]\}\) be a \(d\)-dimensional Brownian motion. Denoting by \(p^ d_ \varepsilon\) the probability density of \(W_ \varepsilon\), the authors derive the representation of \[ \alpha_ \varepsilon (x) = \int^ 1_ 0 \int^ 1_ 0 p^ d_ \varepsilon (W_ t - W_ s - x) dsdt, \qquad x \in R^ d,\;\varepsilon > 0, \] by series of multiple Wiener-Itô integrals for any dimension \(d\), and from this they deduce that \(\alpha(x) = \lim_{\varepsilon \downarrow 0} \alpha_ \varepsilon (x)\) can be considered as a well-defined object in the Sobolev space \(D^{2,\gamma}\) over the canonical Wiener space of any order \(\gamma < (4-d)/2\) if \(x \neq 0\). From here they rederive the existence of the double intersection local time for \(d = 2,3\), shown by J. Rosen [Ann. Probab. 14, 1245-1251 (1986; Zbl 0617.60079), Stochastic Processes Appl. 23, 131-141 (1986; Zbl 0612.60070), and in: Séminaire de probabilités XX, Lect. Notes Math. 1204, 515-531 (1986; Zbl 0611.60065)]; for \(d\geq 4\) the results are new and could be compared with the smoothed versions considered by M. Yor [in: Séminaire de probabilités XIX, Lect. Notes Math. 1123, 332-349 (1985; Zbl 0563.60073)].
For \(x \to 0\), \(\alpha(x)\) does not converge in any Sobolev space, \(\alpha\) has to be renormalised by a suitable deterministic function in order to get the convergence. The difficulty consists in a control of the norms of the projections on the eigenspaces of the Ornstein-Uhlenbeck operator. In case \(d = 2\), S. R. S.Varadhan’s renormalization (1969) is improved, in case \(d = 3\), versions of M. Yor’s renormalization (loc. cit.) turn up, and also the case \(d \geq 4\) is dealt with.
Reviewer: R.Buckdahn (Brest)

MSC:

60H05 Stochastic integrals
60H07 Stochastic calculus of variations and the Malliavin calculus
60J55 Local time and additive functionals
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