map $(\mu,e) \mapsto \overline m_\mu(e)$ is continuous on $(0,\infty) \times \partial B_1$.

Proposition \[p.planarquantitative\] is a consequence of the results in Sections \[s.fluctuations\] and \[s.means\] and its proof comes near the end of Section \[s.means\].

Using the result of Proposition \[p.planarquantitative\], we can identify $\overline H$.

We define the effective Hamiltonian $\overline H:{\mathbb{R}^d}\to {\mathbb{R}}$ by setting $\overline H(0) := 0$ and, for every $t>0$ and $e\in\partial B_1$, $$\overline H(te):= \inf\left\{ \mu>0 \,:\, \overline m_\mu(e) > t \right\}.$$

It is immediate that $\overline H:{\mathbb{R}^d}\to {\mathbb{R}}$ is continuous and coercive and, since $\overline m_\mu(e)$ is increasing in $\mu$, its sublevel sets are star-shaped with respect to the origin. We do not expect $\overline H$ to be positively homogeneous, in general, unless $p=1$ or ${\mathbb{P}}\left[ A\equiv 0 \right]=1$, due to the interaction between the diffusion and the Hamiltonian.

The quantitative error estimates for the time-dependent initial-value problem require an explicit Hölder estimate for $\overline H$, which is proved at the end of Section \[s.correctors\].

\[p.Hbarprops\] For every $L\geq 1$ and $\alpha \in \left( 0,\tfrac 27 \right)$, there exists a constant $C=C({\mathrm{data}},L,\alpha)\geq 1$ such that, for every $\xi,\eta \in \overline B_L$, $$\label{e.Hbarpgrowth1}
c_0 \left| \xi \right|^p \leq \overline H(\xi) \leq C_0 \left| \xi \right|^p$$ 