  In this paper we provide an analysis of a mean first passage time problem of
a random walker subject to a bi-variate $\alpha$-stable L\'evy type noise from
a 2-dimensional disk. For an appropriate choice of parameters the mean first
passage time reveals non-trivial, non-monotonous dependence on the stability
index $\alpha$ describing jumps' length asymptotics both for spherical and
Cartesian L\'evy flights. Finally, we study escape from $d$-dimensional
hyper-sphere showing that $d$-dimensional escape process can be used to
discriminate between various types of multi-variate $\alpha$-stable noises,
especially spherical and Cartesian L\'evy flights.
