We derive a new class of non-linear expectations from first-principles deterministic chaotic dynamics. The homogenization of the system's skew-adjoint microscopic generator is achieved using the spectral theory of transfer operators for uniformly hyperbolic flows. We prove convergence in the viscosity sense to a macroscopic evolution governed by a fully non-linear Hamilton-Jacobi-Bellman (HJB) equation. Our central result establishes that the HJB Hamiltonian possesses a rigid structure: affine in the Hessian but demonstrably non-convex in the gradient. This defines a new $θ$-expectation and constructively establishes a class of non-convex stochastic control problems fundamentally outside the sub-additive framework of G-expectations.

翻译：我们从第一性原理的确定性混沌动力学出发，推导出一类新的非线性期望。通过运用一致双曲流的转移算子谱理论，实现了系统斜伴随微观生成元的均匀化。我们证明了在粘性意义下收敛于由完全非线性Hamilton-Jacobi-Bellman（HJB）方程主导的宏观演化。我们的核心结果表明，HJB哈密顿量具有刚性结构：在Hessian矩阵中是仿射的，但在梯度上被证明是非凸的。这定义了一种新的θ-期望，并构造性地建立了一类非凸随机控制问题，其从根本上超出了G-期望的次可加性框架。