On the minimal integral energy of majorants of the Wiener process

We consider the asymptotic behavior (over long time intervals) of the minimal integral energy $$\mathrm{\mid }h{\mathrm{\mid }}_T^{\psi }={\int }_0^T\psi (h^{\mathrm{\prime }}(t))\mathrm{d}t$$ of majorants of the Wiener process $W(\cdot )$ satisfying the constraints $h(0)=r$, $h(t)\ge W(t)$ for $0\le t\le T$. The results significantly generalize previous asymptotic estimates obtained for the case of kinetic energy $\psi (u)=u^2$, revealing that this case, where the minimal energy grows logarithmically, is a critical one, lying between two different asymptotic regimes.