Limiting Behavior of Non-Autonomous Stochastic Reversible Selkov Lattice Systems Driven by Locally Lipschitz Lévy Noises
Guofu Li, Jianxin Wu, Yunshun Wu
TL;DR
The paper addresses the long-time behavior of infinite-dimensional reversible Selkov lattice systems on $\mathbb{Z}$ driven by locally Lipschitz Lévy noises, formulating a continuous non-autonomous dynamical system on the probability-measure space and proving the existence of a unique pullback measure attractor, which becomes periodic under periodic forcing. It develops uniform moment and tail estimates to overcome the lack of compactness in the lattice setting and establishes pullback asymptotic compactness, enabling the construction of a $\mathcal{D}$-pullback measure attractor $\mathcal{A}(\tau)=\omega(K,\tau)$. The work also proves the upper semicontinuity of the attractors as the noise intensities vanish, providing a robust stability result in the limit of small stochastic perturbations. Numerically, a one-dimensional ODE with Lévy noise demonstrates convergence of stochastic trajectories to the deterministic counterpart as noise vanishes, illustrating the theoretical attractor convergence. Overall, the study extends measure-attractor theory to non-autonomous, jump-driven lattice systems and highlights the role of uniform tail control in achieving asymptotic compactness in infinite dimensions.
Abstract
This work investigates the long-term distributional behavior of the reversible Selkov lattice systems defined on the set $\mathbb{Z}$ and driven by locally Lipschitz \emph{Lévy noises}, which possess two pairs of oppositely signed nonlinear terms and whose nonlinear couplings can grow polynomially with any order $p \geq 1$. Firstly, based on the global-in-time well-posedness in $L^{2}(Ω, \ell^2 \times \ell^2)$, we define a \emph{continuous} non-autonomous dynamical system (NDS) on the metric space $(\mathcal{P}_{2}(\ell^2 \times \ell^2), d_{\mathcal{P}(\ell^2 \times \ell^2)})$, where $d_{\mathcal{P}(\ell^2 \times \ell^2)}$ is the dual-Lipschitz distance on $\mathcal{P}(\ell^2 \times \ell^2)$, the space of probability measures on $\ell^2 \times \ell^2$. Specifically, we establish that this non-autonomous dynamical system admits a unique pullback measure attractor, characterized via measure-valued complete solutions and orbits in the sense of Wang (DOI.org/10.1016/j.jde.2012.05.015). Moreover, when the deterministic external forcing terms are periodic in time, we demonstrate that the pullback measure attractors are also periodic. We also study the upper semicontinuity of pullback measure attractors as $(ε_1, ε_2, γ_1, γ_2) \rightarrow (0, 0, 0, 0)$. The main difficulty in proving the pullback asymptotic compactness of the NDS in $(\mathcal{P}_{2}(\ell^2 \times \ell^2), d_{\mathcal{P}(\ell^2 \times \ell^2)})$ is caused by the lack of compactness in infinite-dimensional lattice systems, which is overcome by using uniform tail-ends estimates. And the inherent structure of the Selkov system precludes the possibility of any unidirectional dissipative influence arising from the interaction between the two coupled equations, thereby obstructing the emergence of a dominant energy-dissipation mechanism along a single directional pathway.