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Local times on rays for a class of planar Lévy processes. (English) Zbl 0807.60071

Summary: Let \(X\), \(Y\) be independent Lévy processes on the real line. Assume \(X\) and \(Y\) admit Lebesgue measure as a reference measure, that \({\mathbf P}^ 0 (X_ t > 0) = c\) for all \(t > 0\) (or the weaker condition \({\mathbf P}^ 0 (X_ t > 0) \to c\) as \(t \to \infty\)) and that \(Y_ t\) has a local time at points. We investigate the distribution of the local time \(L_ t\) of \((X, Y)\) on the positive \(x\)-axis. It turns out that, under the first hypothesis (which is in particular satisfied by planar Brownian motion), if \(T\) is an independent exponential time, then the ratio of \(L_ T\) to \(\ell_ T\), the local time on the entire \(x\) axis, is (generalized) arc-sine and independent of \(\ell_ T\), and \(L_ T\) has a gamma distribution. We obtain then expressions for the distribution of \(L_ t\). In the case of Brownian motion, the formula involves parabolic cylinder functions. Under the weaker condition mentioned above, together with mild secondary hypotheses, we obtain an expression for the asymptotic distribution of \(L_ t\) for large \(t\).

MSC:

60J55 Local time and additive functionals
60J60 Diffusion processes
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