  The time evolution of random variables with L\'evy statistics has the ability
to develop jumps, displaying very different behaviors from continuously
fluctuating cases. Such patterns appear in an ever broadening range of examples
including random lasers, non-Gaussian kinetics or foraging strategies. The
penalizing or reinforcing effect of the environment, however, has been little
explored so far. We report a new phenomenon which manifests as a qualitative
transition in the spatial decay behavior of the stationary measure of a jump
process under an external potential, occurring on a combined change in the
characteristics of the process and the lowest eigenvalue resulting from the
effect of the potential. This also provides insight into the fundamental
question of what is the mechanism of the spatial decay of a ground state.
