by stationarity can be written as a deterministic function of the form $f(x\cdot e)$, must therefore be close to a plane, that is, $f$ is close to a linear function. For the last step we think of the distance to the boundary plane as “time" and consider the “semigroup" generated by the “flow" and use a maximum principle argument. The fact that $\partial {\mathcal{H}}_e^+$ is unbounded and the fluctuations estimate is not uniform raises another difficulty which is overcome by a new “approximate finite speed of propagation" property presented in Section \[s.propagation\].

Precise assumptions and statement of homogenization {#s.statements}
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We begin with the structural conditions on the coefficients in  before giving the probabilistic formulation of the problem. Throughout that paper, we fix an exponent $p\geq 1$, a dimension $d\in{\mathbb{N}}_*$, a positive integer $n\in{\mathbb{N}}_*$, constants $0<c_0 \leq C_0 <\infty$ and a modulus $\rho \in C([1,\infty))$ satisfying $$\label{}
\lim_{R\to \infty} \rho(R) = \infty.$$ We also fix parameters $\theta,\kappa >0$. It is convenient to set $$\label{}
{\mathrm{data}}:= \left(d,p,n,c_0,C_0,\rho,\theta,\kappa \right).$$

We consider diffusion matrices $A: \partial B_1 \times {\mathbb{R}^d}\to {\mathbb{R}}^{d\times d}$ which take the form $$\label{e.Asigma}
A = \frac12 \sigma\sigma^T, \qquad \mbox{where $\sigma\in C^{1}(\partial B_1 \times{\mathbb{R}^d}; \, {\mathbb{R}}^{d\times n})$}$$ is a matrix-valued function 