Stochastic Lie Algebra Decoupling in Reflexive Banach Spaces: Normal Forms for Noisy Quadratic Systems in Infinite Dimensions

We extend the stochastic Lie algebra decoupling framework for Itô-type stochastic differential equations (SDEs) from separable Hilbert spaces to reflexive Banach spaces with a countable Schauder basis, considering equations of the form du = [Au + B(u, u)]dt + σ(u)dW , where A is a possibly unbounded linear operator, B is a quadratic bilinear form, and σ(u) represents multiplicative noise driven by an infinite-dimensional Wiener process W. Leveraging reflexivity for well-defined adjoints, weak compactness, and stochastic integrals, we establish stochastic resonant conditions using adjoint representations and prove solvability of the stochastic homological equation under non-resonance assumptions, yielding normal forms that eliminate non-resonant quadratic-noise terms. This generalization addresses domain issues in non-Hilbert settings, such as L p spaces for 1 < p < ∞, and extends to higher-order normal forms with convergence in Gevrey classes via stochastic involutive PDE theory and the Cartan-Kähler theorem, mitigating small divisors through spectral gaps in stochastic averages. Applications to noisy quantum many-body systems (e.g., stochastic Hartree and Hartree-Fock equations in Sobolev embeddings), turbulent fluid dynamics in reflexive spaces with noise, and stochastic topological insulators demonstrate reduced computational complexity, mode decoupling under uncertainty, and preservation of stochastic invariants like reversibility and martingale properties. Numerical validations on models like the stochastic Bose-Hubbard system underscore the approach’s efficacy in handling noise-induced resonances. This work broadens finite-dimensional stochastic Lie theory to unbounded operators in reflexive Banach spaces, offering new insights into stochastic resonances, stability, and emergent behaviors in complex noisy infinite-dimensional systems.