On the minimal integral energy of majorants of the Wiener process

TL;DR

This paper investigates the long-term asymptotic behavior of the minimal integral energy of functions that majorize a Wiener process, revealing a critical case where energy growth is logarithmic, bridging two distinct regimes.

Contribution

It generalizes previous results by analyzing a broad class of energy functions, showing the critical nature of the quadratic case in the asymptotic growth of minimal energy.

Findings

Quadratic energy case exhibits logarithmic growth.

General energy functions reveal a transition between asymptotic regimes.

Results extend understanding of Wiener process majorants over long intervals.

Abstract

We consider the asymptotic behavior (over long time intervals) of the minimal integral energy \[ |h|_T^\psi = \int_0^T \psi(h^\prime(t)) \, \mathrm{d}t \] of majorants of the Wiener process $ W(\cdot) $ satisfying the constraints $ h(0) = r $, $ h(t) \geq W(t) $ for $ 0 \leq t \leq 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.