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Cooling down Levy flights

I. Pavlyukevich
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Abstract

Let L(t) be a Levy flights process with a stability index α\in(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+σ(t)dL(t) describes an evolution of a Levy particle of an `instant temperature' σ(t) in an external force field. The temperature is supposed to decrease polynomially fast, i.e. σ(t)\approx t^{-θ} for some θ>0. We discover two different cooling regimes. If θ<1/α(slow cooling), the jump diffusion Z(t) has a non-trivial limiting distribution as t\to \infty, which is concentrated at the potential's local minima. If θ>1/α(fast cooling) the Levy particle gets trapped in one of the potential wells.

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I. Pavlyukevich. 2007-01-26. Cooling down Levy flights. https://doi.org/10.1088/1751-8113%2F40%2F41%2F003

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