论文标题
汉密尔顿 - 雅各比 - 贝尔曼方程的随机均匀化，连续渗透簇
Stochastic homogenization for Hamilton-Jacobi-Bellman equations on continuum percolation clusters
论文作者
论文摘要
我们证明了随机汉密尔顿 - 雅各比 - 贝尔曼（HJB）方程在连续渗透簇上，几乎肯定是W.R.T.原点属于连续体的无限组成部分时的环境定律。在这里，粘度术语带有退化矩阵，哈密顿量是凸和强制W.R.T.退化矩阵和基础环境是非椭圆形的，其定律是非平稳的W.R.T.翻译组。我们不假定渗透集群内部均匀的椭圆度，也不假设对渗透模型的任何有限范围依赖性（I.I.D.）假设，而有效的哈密顿人则承认反映了渗透的某些关键特性的变异公式。该证明的灵感来自一种Kosygina-Rezakhanlou-varadhan [KRV06]，该方法是为HJB方程开发的，该方程在固定的，成猫和椭圆的随机环境中具有恒定的粘度和均匀的强制性Hamiltonian。在非平稳和非胸腔设置中，我们利用了基础汉密尔顿和相对熵结构的强化特性（在任何框架中都是HJB的固有特性），并利用连续性渗透的随机几何形状。
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.
