  We analyze a specific class of random systems that are driven by a symmetric
L\'{e}vy stable noise. In view of the L\'{e}vy noise sensitivity to the
confining "potential landscape" where jumps take place (in other words, to
environmental inhomogeneities), the pertinent random motion asymptotically sets
down at the Boltzmann-type equilibrium, represented by a probability density
function (pdf) $\rho_*(x) \sim \exp [-\Phi (x)]$. Since there is no Langevin
representation of the dynamics in question, our main goal here is to establish
the appropriate path-wise description of the underlying jump-type process and
next infer the $\rho (x,t)$ dynamics directly from the random paths statistics.
A priori given data are jump transition rates entering the master equation for
$\rho (x,t)$ and its target pdf $\rho_*(x)$. We use numerical methods and
construct a suitable modification of the Gillespie algorithm, originally
invented in the chemical kinetics context. The generated sample trajectories
show up a qualitative typicality, e.g. they display structural features of
jumping paths (predominance of small vs large jumps) specific to particular
stability indices $\mu \in (0,2)$.
