Strong and weak convergence rates for slow-fast system driven by multiplicative L\'{e}vy noises

TL;DR

This paper investigates the convergence rates of slow-fast stochastic systems driven by multiplicative Lévy noises, providing new ergodicity results and explicit formulas for tangent maps, with implications for stochastic analysis.

Contribution

It introduces novel methods to establish ergodicity and convergence rates for systems with multiplicative Lévy noise, extending existing additive noise results.

Findings

Established exponential ergodicity using coupling and spatial periodic methods

Derived optimal strong convergence rate of order 1 - 1/α₂

Obtained explicit formulas for tangent maps and Jacobian determinants

Abstract

This paper establishes strong and weak convergence rates for slow-fast systems driven by $\alpha$-stable processes with jump coefficients. Unlike existing studies on multiscale systems driven by additive L\'{e}vy 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 H\"{o}lder regularity of the time-dependent coefficients of the slow process, we can yield an optimal strong convergence rate of order $1-\frac{1}{\alpha_{2}}$ 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.