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 \[ |h|_T^ψ= ∫_0^T ψ(h^(t)) \, dt \] of majorants of the Wiener process W(·) satisfying the constraints $ h(0) = r $, h(t) ≥ W(t) for 0 ≤ t ≤ T. The results significantly generalize previous asymptotic estimates obtained for the case of kinetic energy ψ(u) = u², revealing that this case, where the minimal energy grows logarithmically, is a critical one, lying between two different asymptotic regimes.