Variational Principles on Geometric Rough Paths and the L\'{e}vy Area Correction

TL;DR

This paper investigates how the Lévý area correction affects the invariant measure of stochastic rigid body dynamics on geometric rough paths, revealing a deterministic torque that shifts the probability distribution.

Contribution

It introduces a novel analysis of the Lévý area correction's impact on invariant measures and demonstrates its effect as a deterministic torque in the dynamics.

Findings

Lévý area correction adds a deterministic torque to the rigid body equations.

The correction shifts the invariant measure's center by modifying the Hamiltonian.

Numerical results confirm the theoretical predictions about the measure shift.

Abstract

In this paper, we describe two effects of the L\'evy area correction on the invariant measure of stochastic rigid body dynamics on geometric rough paths. From the viewpoint of dynamics, the L\'evy area correction introduces an additional deterministic torque into the rigid body motion equation on geometric rough paths. When the dynamics is driven by coloured noise, and for rigid body dynamics with double-bracket dissipation, theoretical and numerical results show that this additional deterministic torque shifts the centre of the probability distribution function by shifting the Hamiltonian function in the exponent of the Gibbsian invariant measure.