L\'evy area with a drift as a renormalization limit of Markov chains on periodic graphs

TL;DR

This paper investigates the intrinsic drift in the LévY area of Brownian motion within rough path topology, constructing it via renormalization limits of Markov chains on periodic graphs and providing explicit formulas and examples.

Contribution

It introduces a novel construction of the intrinsic drift in the LévY area using renormalization limits, with explicit formulas and illustrative examples.

Findings

Explicit formula for the intrinsic drift of the LévY area.

Construction of the drift via renormalization limits of Markov chains.

Multiple examples demonstrating the computation of the drift.

Abstract

A careful look at rough path topology applied to Brownian motion reveals new possible properties of the well-known L\'evy area, in particular the presence of an intrinsic drift of this area. Using renormalization limit of Markov chains on periodic graphs, we present a construction of such a non-trivial drift and give an explicit formula for it. Several examples for which explicit computations are made are included.