We develop a quenched thermodynamic formalism for a wide class of random maps with non-uniform expansion, where no Markov structure, no uniformly bounded degree or the existence of some expanding dynamics is required. We prove that every measurable and fibered \(C^1\)-potential at high temperature admits a unique equilibrium state which satisfies a weak Gibbs property, and has exponential decay of correlations. The arguments combine a functional analytic approach for the decay of correlations (using Birkhoff cone methods) and Carathéodory-type structures to describe the relative pressure of not necessary compact invariant sets in random dynamical systems. We establish also a variational principle for the relative pressure of random dynamical systems.

我们针对一类具有非均匀扩展的随机映射开发了一个淬火的热力学形式论，其中不需要马尔可夫结构，一致界度或存在一些扩展动力学。我们证明，高温下每个可测量的纤维化\（C ^ 1 \）势均具有满足弱Gibbs性质的独特平衡状态，并且具有指数衰减的相关性。这些论点结合了一种用于相关性衰减的函数分析方法（使用Birkhoff锥方法）和Carathéodory型结构，以描述随机动力系统中不必要的紧不变集的相对压力。我们还为随机动力系统的相对压力建立了变分原理。