Cooling down Levy flights

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.

References in corpus (6)

• Levy Flights in External Force Fields: Langevin and Fractional Fokker-Planck Equations, and their Solutions
• Anomalous jumping in a double-well potential
• Lévy flights in a steep potential well
• Levy flights from a continuous-time process
• Levy Flights in External Force Fields: From Models to Equations
• Escape driven by $\alpha$-stable white noises

Cited by in corpus (6)

• Levy Flight Superdiffusion: An Introduction
• First exit times for Lévy-driven diffusions with exponentially light jumps
• Lévy Ratchet in a Weak Noise Limit: Theory and Simulation
• Hausdorff Dimension, Heavy Tails, and Generalization in Neural Networks
• Bounding the first exit from the basin: Independence Times and Finite-Time Basin Stability
• How to simulate Lévy flights in a steep potential: An explicit splitting numerical scheme