generated by the family of maps $\Omega \to {\mathbb{R}}^{d\times n} \times {\mathbb{R}}$,} \\ 
(\sigma,H) \mapsto (\sigma(e,x),H(\xi,x)), \quad\mbox{where} \ e\in\partial B_1, \ \xi\in{\mathbb{R}^d}\ \mbox{and} \ x\in U.\end{gathered}$$ The largest of these we denote by ${\mathcal{F}}:={\mathcal{F}}({\mathbb{R}^d})$. The interpretation of ${\mathcal{F}}(U)$ is that it is the $\sigma$–algebra containing “all of the information which can be obtained by observing the coefficients restricted to $U$."

Throughout the paper, we consider a fixed probability measure ${\mathbb{P}}$ on $(\Omega,{\mathcal{F}})$ which satisfies the following two conditions:

1.  ${\mathbb{P}}$ is stationary: for every $y\in{\mathbb{R}^d}$, we have that $${\mathbb{P}}= {\mathbb{P}}\circ T_y,$$ where $T_y:\Omega\to \Omega$ acts on $\Omega$ by translation in $y$, i.e., $$T_y(\sigma,H):= ((e,z)\mapsto \sigma(e,z+y), (\xi,z)\mapsto H(\xi,z+y)).$$

2.  ${\mathbb{P}}$ satisfies a unit range of dependence: for all Borel subsets $U,V\subseteq {\mathbb{R}^d}$ such that $\operatorname{dist}(U,V) \geq 1$, we have $$\mbox{${\mathcal{F}}(U)$ and ${\mathcal{F}}(V)$ are ${\mathbb{P}}$--independent.}$$

Throughout the paper, all differential inequalities are to be understood in the viscosity sense. Since the quasilinear diffusions we consider have singularities at $\xi=0$, and for the readers’ convenience, we recall the appropriate definitions in Section \[s.notation\] below.

The main result of the paper is Theorem \[t.main\], stated at the beginning of Section \[s.outline\]. Here we present a consequence of it which is simpler, more qualitative and 