Getoor, R. K.; Sharpe, M. J. Local times on rays for a class of planar Lévy processes. (English) Zbl 0807.60071 J. Theor. Probab. 7, No. 4, 799-811 (1994). 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 \(t\). Cited in 2 Documents MSC: 60J55 Local time and additive functionals 60J60 Diffusion processes Keywords:arc-sine law; stable process; local time; Lévy processes; planar Brownian motion; asymptotic distribution PDFBibTeX XMLCite \textit{R. K. Getoor} and \textit{M. J. Sharpe}, J. Theor. Probab. 7, No. 4, 799--811 (1994; Zbl 0807.60071) Full Text: DOI References: [1] Getoor, R. K., and Sharpe, M. J. (1994). On the arc-sine laws for Lévy processes,J. Appl. Prob.31, 76-89. · Zbl 0802.60070 · doi:10.2307/3215236 [2] Blumenthal, R. M., and Getoor, R. K. (1968).Markov Processes and Potential Theory, Academic Press, San Diego. · Zbl 0169.49204 [3] Getoor, R. K., and Sharpe, M. J. (1973). Last exit times and additive functions.Ann. Prob.1, 550-569. · Zbl 0324.60062 · doi:10.1214/aop/1176996885 [4] Bateman, H., Erdélyi, A. et al. (1954a).Tables of Integral Transforms, Vol. I, McGraw-Hill, New York. · Zbl 0055.36401 [5] Bateman, H., Erdélyi, A. et al. (1954b).Transcendental Functions, Vol. II, McGraw-Hill, New York. · Zbl 0143.29202 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.