The codimension of the zeros of a stable process in random scenery. (English) Zbl 1040.60087

Azéma, J. (ed.) et al., 37th seminar on probability. Berlin: Springer (ISBN 3-540-20520-9/pbk). Lect. Notes Math. 1832, 236-245 (2003).
Author’s abstract: We show that for any \(\alpha\in(1, 2]\), the (stochastic) codimension of the zeros of an \(\alpha\)-stable process in random scenery is identically \(1-(2\alpha)^{-1}\). As an immediate consequence, we deduce that the Hausdorff dimension of the zeros of the latter process is almost surely equal to \((2\alpha)^{-1}\). This solves Conjecture 5.2 of the author and T. M. Lewis [in: Seminar on stochastic analysis, random fields and applications. Prog. Probab. 45, 201–210 (1999; Zbl 0943.60081)], thereby refining a computation of Y. Xiao [Acta Sci. Math. 65, 385–395 (1999; Zbl 0937.60030)].
For the entire collection see [Zbl 1027.00025].


60K37 Processes in random environments