Cooling down Lévy flights

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