Two types of averaging principles for multiplicative stochastic Korteweg-de Vries equations with fast oscillations
Abstract
In this paper, we study averaging principles for stochastic KdV equations perturbed by a fast oscillating term generated by a stochastic reaction-diffusion equation in Hs(ℝ), $ s\geq 0$. When dealing with multilinear estimates of the nonlinear terms in the stochastic KdV type equation, two kinds of workspace are both effective. The classical workspace can help us prove the averaging principle in the Lω2 sense, but it has more restrictions on the noise kernel and the conservation structures. When we use a new workspace without homogeneous low frequency restrictions, only the regularity conditions and the Lx2 conservation structure is needed. Correspondingly, because there is no moment estimate of the solution, we can only prove the perturbed equations converge to an averaging equation in the probability sense. Our works may have important reference value, because it not only provides a new perspective to handle dynamical problems of stochastic dispersive model, but also compare it with the classical method.
Related
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BibTeX
@misc{chen2026types,
title = {{Two types of averaging principles for multiplicative stochastic Korteweg-de Vries equations with fast oscillations}},
author = {Jie Chen and Fan Gu and Xianliang Zhao},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01051},
eprint = {2611.01051},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01051},
}
RIS
TY - UNPB
TI - Two types of averaging principles for multiplicative stochastic Korteweg-de Vries equations with fast oscillations
AU - Jie Chen
AU - Fan Gu
AU - Xianliang Zhao
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01051
UR - https://arxiv.org/abs/2611.01051
AB - In this paper, we study averaging principles for stochastic KdV equations perturbed by a fast oscillating term generated by a stochastic reaction-diffusion equation in H^s(ℝ), $ s\geq 0$. When dealing with multilinear estimates of the nonlinear terms in the stochastic KdV type equation, two kinds of workspace are both effective. The classical workspace can help us prove the averaging principle in the L_ω^2 sense, but it has more restrictions on the noise kernel and the conservation structures. When we use a new workspace without homogeneous low frequency restrictions, only the regularity conditions and the L_x^2 conservation structure is needed. Correspondingly, because there is no moment estimate of the solution, we can only prove the perturbed equations converge to an averaging equation in the probability sense. Our works may have important reference value, because it not only provides a new perspective to handle dynamical problems of stochastic dispersive model, but also compare it with the classical method.
ER -