L��vy stable (jump-type) processes are examples of intrinsically nonlocal random motions. This property becomes a serious obstacle if one attempts to model conditions under which a particular L��vy process may be subject to physically implementable manipulations, whose ultimate goal is to confine the random motion in a spatially finite, possibly mesoscopic trap. We analyze thisissue for an exemplary case of the Cauchy process in a finiteinterval. Qualitatively, our observations extend to general jump-type processes that are driven by non-gaussian noises, classified by the integral part of the L��vy-Khintchine formula.For clarity of arguments we discuss, as a reference model, the classic case of the Brownian motion in the interval.

11 pp, 7 figures. In this version, minor correction next to Eq. (7)