Strong and weak approximation for a two-time-scale stochastic quasi-geostrophic flow equation driven by Lévy processes

Pin Wang; Pin Wang

doi:10.3934/math.2026090

AIMS Mathematics

2026, Volume 11, Issue 1: 2227-2254. doi: 10.3934/math.2026090

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Research article

Strong and weak approximation for a two-time-scale stochastic quasi-geostrophic flow equation driven by Lévy processes

Pin Wang ^,

School of Mathematics and Statistics, Chongqing University of Posts and Telecommunications, Chongqing, 400065, China

Received: 16 October 2025 Revised: 23 December 2025 Accepted: 04 January 2026 Published: 23 January 2026
MSC : 34C29, 35R60, 37A25, 60H15

This work investigates the strong and weak approximation for a stochastic quasi-geostrophic flow equation with two time scales, where the slow component is coupled with a fast oscillation governed by a stochastic reaction-diffusion equation, and both are driven by Lévy noises. Employing Khasminskii's time discretization, we first prove that the slow component of the slow-fast system converges to the solution of the averaged equation in a strong sense with the help of an auxiliary process in small subintervals. Based on an asymptotic expansion of solutions for the Kolmogorov equation associated with the slow-fast system through a discontinuous path, we then decompose the weak solution with respect to the small parameter. By means of the components being determined recursively, we further establish the weak convergence from the original to the averaged dynamics.
- stochastic quasi-geostrophic flow equation,
- fast oscillation,
- averaging principle,
- weak and strong convergence,
- asymptotic expansion
Citation: Pin Wang. Strong and weak approximation for a two-time-scale stochastic quasi-geostrophic flow equation driven by Lévy processes[J]. AIMS Mathematics, 2026, 11(1): 2227-2254. doi: 10.3934/math.2026090

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Abstract

This work investigates the strong and weak approximation for a stochastic quasi-geostrophic flow equation with two time scales, where the slow component is coupled with a fast oscillation governed by a stochastic reaction-diffusion equation, and both are driven by Lévy noises. Employing Khasminskii's time discretization, we first prove that the slow component of the slow-fast system converges to the solution of the averaged equation in a strong sense with the help of an auxiliary process in small subintervals. Based on an asymptotic expansion of solutions for the Kolmogorov equation associated with the slow-fast system through a discontinuous path, we then decompose the weak solution with respect to the small parameter. By means of the components being determined recursively, we further establish the weak convergence from the original to the averaged dynamics.

References

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