  We prove homogenization properties of random 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.) assumption on the percolation
models and the effective Hamiltonian admits a variational formula which
reflects some key properties of percolation. The proof is inspired by a method
of Kosygina-Rezakhanlou-Varadhan [KRV06] developed for the case of HJB
equations with constant viscosity and uniformly coercive Hamiltonian in a
stationary, ergodic and elliptic random environment. In the non-stationary and
non-elliptic set up, we leverage the coercivity property of the underlying
Hamiltonian as well as a relative entropy structure (both being intrinsic
properties of HJB, in any framework) and make use of the random geometry of
continuum percolation.
