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Joint continuity of renormalized intersection local times. (English) Zbl 0867.60049

Let \(X\) be a planar symmetric stable process with index \(\beta\in(0,2]\) and \(k\geq 2\) an integer such that \((2k- 1)(2-\beta)<2\). For each \(t>0\), the \(k\)-fold intersection local time \(\alpha_k(x_2,\dots,x_k,t)\) is defined as the density at the point \((x_2,\dots,x_k)\in({\mathbf R}^2-\{0\})^{k-1}\) of the measure given by \[ \int\cdots\int_{\{0\leq t_1\leq\cdots\leq t_k\leq t\}}\text{\textbf{1}}_{\{(X_{t_2}- X_{t_1},\dots,X_{t_k}- X_{t_{k-1}})\in\bullet\}}dt_1\cdots dt_k. \] It is proven that the mapping \(\alpha_k\) is continuous on \(({\mathbf R}^2-\{0\})^{k-1}\times[0,\infty)\) and can be renormalized in such a way that the renormalized intersection local time has a continuous extension to \(({\mathbf R}^2)^{k-1}\times [0,\infty)\).
Reviewer: J.Bertoin (Paris)

MSC:

60J65 Brownian motion
60J55 Local time and additive functionals
60H05 Stochastic integrals
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