Cooling down Levy flights

TL;DR

This paper investigates how Levy flights in a potential field behave under polynomially decreasing temperature, revealing two regimes: one with a limiting distribution at minima and another with trapping in a well.

Contribution

It introduces a model of Levy particles with decreasing temperature and characterizes the different long-term behaviors depending on the cooling rate.

Findings

Slow cooling leads to a non-trivial limiting distribution at local minima.

Fast cooling causes the Levy particle to become trapped in a potential well.

Two distinct regimes are identified based on the cooling rate relative to the stability index.

Abstract

Let L(t) be a Levy flights process with a stability index \alpha\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+\sigma(t)dL(t) describes an evolution of a Levy particle of an `instant temperature' \sigma(t) in an external force field. The temperature is supposed to decrease polynomially fast, i.e. \sigma(t)\approx t^{-\theta} for some \theta>0. We discover two different cooling regimes. If \theta<1/\alpha (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 \theta>1/\alpha (fast cooling) the Levy particle gets trapped in one of the potential wells.