Stochastic homogenization for Hamilton-Jacobi-Bellman equations on continuum percolation clusters

TL;DR

This paper establishes homogenization results for stochastic Hamilton-Jacobi-Bellman equations on continuum percolation clusters, addressing non-elliptic, non-stationary environments without uniform ellipticity or finite-range dependence.

Contribution

It extends homogenization theory to non-elliptic, non-stationary continuum percolation environments for HJB equations, using a variational formula and entropy methods.

Findings

Homogenization holds almost surely on the infinite cluster.

Effective Hamiltonian admits a variational formula reflecting percolation properties.

Method adapts to non-elliptic, non-stationary settings without uniform ellipticity.

Abstract

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.