Cooling down Levy flights

Let L(t) be a Levy flights process with a stability index alpha is an element of (0, 2), and U be an external multi-well potential. A jump diffusion Z satisfying a stochastic differential equation dZ(t) =-U'(Z(t-)) dt + sigma(t) dL(t) describes an evolution of a Levy particle of an 'instant scale' sigma(t) in an external force field. The scale is supposed to decrease polynomially fast, i.e. sigma(t) approximate to t(-theta) for some theta > 0. We discover two different decrease regimes. If theta < 1/alpha (slow cooling), the jump diffusion Z(t) has a non-trivial limiting distribution as t ->infinity, which is concentrated at the potential's local minima. If theta > 1/alpha (fast cooling), the Levy particle gets trapped in one of the potential wells.