A local variational principle for random bundle transformations

We introduce local topological entropy $h_{\text{top}}(T, \mathcal{U})$ and two kinds of local measure-theoretic entropy $h_{\mu}^{(r)-}(T,\mathcal{U})$ and $h_{\mu}^{(r)+}(T,\mathcal{U})$ for random bundle transformations. We derive a variational inequality of random local entropy for $h_{\mu}^{(r)+}(T,\mathcal{U})$. As an application of such relation we prove a local variational principle in random dynamical system.