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Strong and weak convergence rates for slow-fast system driven by multiplicative Lévy noises

Qiu-Chen Yang, Kun Yin

Abstract

This paper establishes strong and weak convergence rates for slow-fast systems driven by $α$-stable processes with jump coefficients. Unlike existing studies on multiscale systems driven by additive Lévy white noise, our model incorporates multiplicative noise, which brings essential challenges in deriving the exponential ergodicity for the frozen process, particularly gradient estimates. We derive exponential ergodicity in two different ways: the coupling method and the spatial periodic method; then the gradient estimate is developed by heat kernel asymptotic expansion. Moreover, under sufficient Hölder regularity of the time-dependent coefficients of the slow process, we can yield an optimal strong convergence rate of order $1-\frac{1}{α_{2}}$ and a weak convergence rate of order 1. Furthermore, explicit formulas for the tangent map between tangent spaces of $S^{d-1}$ as well as its Jacobian determinant are obtained, where the map is induced by a nonlinear immersion.

Strong and weak convergence rates for slow-fast system driven by multiplicative Lévy noises

Abstract

This paper establishes strong and weak convergence rates for slow-fast systems driven by -stable processes with jump coefficients. Unlike existing studies on multiscale systems driven by additive Lévy white noise, our model incorporates multiplicative noise, which brings essential challenges in deriving the exponential ergodicity for the frozen process, particularly gradient estimates. We derive exponential ergodicity in two different ways: the coupling method and the spatial periodic method; then the gradient estimate is developed by heat kernel asymptotic expansion. Moreover, under sufficient Hölder regularity of the time-dependent coefficients of the slow process, we can yield an optimal strong convergence rate of order and a weak convergence rate of order 1. Furthermore, explicit formulas for the tangent map between tangent spaces of as well as its Jacobian determinant are obtained, where the map is induced by a nonlinear immersion.
Paper Structure (20 sections, 249 equations)

This paper contains 20 sections, 249 equations.

Table of Contents

  1. Introduction
  2. Backgrounds
  3. Sketch of this paper
  4. Organization of this paper
  5. Some settings and main results
  6. Notations and assumptions
  7. Main results
  8. Well-posedness and some moment estimates of $(X_{t}^{\varepsilon},Y_{t}^{\varepsilon})$
  9. The frozen equation for \ref{['1.1']}
  10. Exponential decay with respect to $L^p$-Wasserstein distance
  11. Invariant measure of \ref{['4.1']}: general case
  12. Invariant measure of projected frozen equation: spatial periodic case
  13. Moment estimates of $Y_{t}^{ x,y}$
  14. Strong convergence estimates for \ref{['1.1']}
  15. Weak convergence estimates for \ref{['1.1']}
  16. ...and 5 more sections

Theorems & Definitions (20)

  • proof
  • proof
  • proof
  • proof : Proof of Theorem $\ref{['was-exp']}$
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 10 more