Well-posedness, mean attractors and invariant measures of stochastic discrete long-wave-short-wave resonance equations driven by locally Lipschitz nonlinear noise

TL;DR

This paper establishes the well-posedness and long-term statistical properties of stochastic discrete long-wave-short-wave resonance equations with nonlinear noise, using advanced functional analysis techniques.

Contribution

It introduces a novel phase space and proves the existence of mean attractors and invariant measures for these complex stochastic systems.

Findings

Global well-posedness in a higher-order Bochner space

Existence and uniqueness of mean random attractors

Convergence of invariant measures as noise diminishes

Abstract

This paper is devoted to investigating the random dynamics of stochastic discrete long-wave-short-wave resonance equations, which are characterized by the following features: $(1)$ the equations contain locally Lipschitz nonlinear coupling terms $u_mv_m$ and $(B(|u(t)|^2))_m$ for $m\in \mathbb{Z}$; $(2)$ the nonlinear coefficients of noises satisfy local Lipschitz conditions; and $(3)$ the system couples real and complex equations and is infinite-dimensional. These inherent structural properties prevent the analysis from being carried out in a standard Bochner product space of the same order and make it difficult to directly verify the tightness of the distribution family of solutions. To address these challenges, we adopt a higher-order Bochner product space $L^4(\Omega,\ell_c^2)\times L^2(\Omega,\ell^2)$ as the phase space and employ the technique of uniform tail-end estimates. The main results include: establishing the global well-posedness of the nonautonomous stochastic discrete long-wave-short-wave resonance equations driven by nonlinear noise in $L^4(\Omega,\ell_c^2)\times L^2(\Omega,\ell^2)$; based on this, defining the mean random dynamical system and proving the existence and uniqueness of weak $\mathscr{D}$-pullback mean random attractors. When the external forcing terms are independent of time and sample, we investigate the existence of invariant measures for the corresponding autonomous system and examine the limiting behavior of the invariant measure as the noise intensity tends to zero.