  We analyze two-dimensional (2D) random systems driven by a symmetric L\'{e}vy
stable noise which, under the sole influence of external (force) potentials
$\Phi (x) $, asymptotically set down at Boltzmann-type thermal equilibria. Such
behavior is excluded within standard ramifications of the Langevin approach to
L\'{e}vy flights. In the present paper we address the response of L\'{e}vy
noise not to an external conservative force field, but directly to its
potential $\Phi (x)$. We prescribe a priori the target pdf $\rho_*$ in the
Boltzmann form $\sim \exp[- \Phi (x)]$ and next select the L\'evy noise of
interest. Given suitable initial data, this allows to infer a reliable
path-wise approximation to a true (albeit analytically beyond the reach)
solution of the pertinent master equation, with the property $\rho
(x,t)\rightarrow \rho_*(x)$ as time $t$ goes to infinity. We create a suitably
modified version of the time honored Gillespie's algorithm, originally invented
in the chemical kinetics context. A statistical analysis of generated sample
trajectories allows us to infer a surrogate pdf dynamics which consistently
sets down at a pre-defined target pdf. We pay special attention to the response
of the 2D Cauchy noise to an exemplary locally periodic "potential landscape"
$\Phi (x), x\in R^2$.
