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The Diffusion of Passive Tracers in Laminar Shear Flow
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Superdiffusive limits for deterministic fast-slow dynamical systems.

Ilya Chevyrev1, Peter K Friz2,3, Alexey Korepanov4

  • 1School of Mathematics, University of Edinburgh, Edinburgh, EH9 3FD UK.

Probability Theory and Related Fields
|November 13, 2020
PubMed
Summary
This summary is machine-generated.

This study demonstrates the convergence of fast-slow dynamical systems to stochastic differential equations driven by Lévy processes. These findings are applicable to intermittent maps, offering insights into complex system dynamics.

Keywords:
Primary 37A50Secondary 60L20

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Area of Science:

  • Dynamical Systems and Chaos Theory
  • Stochastic Processes and Analysis
  • Mathematical Physics

Background:

  • Fast-slow dynamical systems exhibit complex behaviors.
  • Stochastic differential equations are crucial for modeling random phenomena.
  • Lévy processes generalize Brownian motion to include jumps.

Purpose of the Study:

  • To establish convergence of a deterministic fast-slow system to a stochastic differential equation.
  • To analyze the properties of the resulting stochastic process.
  • To validate the theoretical framework using specific examples.

Main Methods:

  • Analysis of deterministic fast-slow dynamical systems.
  • Convergence proofs using stochastic calculus.
  • Application of Marcus integral definition for stochastic integrals.
  • Verification with Pomeau-Manneville type intermittent maps.

Main Results:

  • Proved convergence of the m-dimensional process to a stochastic differential equation.
  • Identified the driving process as an alpha-stable Lévy process.
  • Demonstrated that assumptions hold for intermittent maps.

Conclusions:

  • The study provides a rigorous mathematical link between deterministic and stochastic models.
  • The findings are relevant for understanding systems with intermittent dynamics.
  • This work contributes to the theory of stochastic processes and dynamical systems.