Antilocality and one-sided antilocality for stable generators on the line (Q1086919)

We study the propagation of zeroes for a test function f with \(Af=0\), where A is the generator of one-dimensional stable processes with index \(\alpha\), \(0<\alpha <1\). It is proved that the zeroes for f propagate along the support of the Lévy measure for the corresponding stable process. The result can be extended for the case \(\alpha\in [1,2)\) via similar arguments. Recently a similar result for two-dimensional case has been obtained, which is now under preparation for publication.