Homogenization of Hamilton-Jacobi-Bellman equations on continuum percolation clusters

We prove homogenization of Hamilton-Jacobi-Bellman (HJB) equations on continuum percolation clusters, almost surely w.r.t. the law of the environment when the origin belongs to the unbounded component in the continuum. Here, the viscosity term carries a degenerate matrix, the Hamiltonian is convex and coercive w.r.t. the degenerate matrix and the underlying environment is non-elliptic and its law is non-stationary w.r.t. the translation group. We do not assume uniform ellipticity inside the percolation cluster, nor any finite-range dependence (i.i.d.) on the environment. In this set up, we develop an approach based on the coercivity property of the Hamiltonian as well as a relative entropy structure (both being intrinsic properties of HJB) and make use of the random geometry of continuum percolation. Joint work with Rodrigo Bazaes (Münster) and Alexander Milke (Berlin).