Large deviation principle for stochastic \(p\)-Laplacian reversible Selkov lattice systems

Abstract

The Selkov system is a classical model for autocatalytic biochemical reactions such as glycolysis and exhibits complex spatio-temporal patterns including self-excitation, oscillations, and spatial chaos.  In this paper, we consider stochastic \(p\)-Laplacian reversible Selkov lattice systems defined on the integer set \(\mathbb{Z}\), which possess two pairs of oppositely signed nonlinear terms and whose nonlinear couplings can grow polynomially with any order \(q \geq 1\), the inherent structure of the 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. In particular, our focus is on studying the asymptotic properties of the \((\psi^\gamma, \phi^\gamma)\) as the noise intensity \(\gamma \to 0\)  for the systems.  Based on the well-posedness and the convergence of the solutions of the controlled system associated with the considered system, by the weak convergence method, we establish the large deviation principle in \( C([0,T], \ell^2 \times \ell^2) \) for such infinite dimensional system. One of the advantages of the weak convergence method that does not rely on uniform exponential probability estimates of solutions. Moreover, a stopping time technique is used to prove \(\lim\limits_{\gamma \to 0} \left( \psi^\gamma_{z^\gamma} - \psi_{z^\gamma}, \phi^\gamma_{z^\gamma} - \phi_{z^\gamma} \right) = 0\) in probability in order to overcome the difficulty caused by the local monotonicity of the nonlinear and diffusion terms. Compared to the classical case of cubic nonlinearity, our results are not only more general in scope but also demonstrate a higher degree of novelty.