Integration and stochastic integration in Gaussian multiplicative chaos

TL;DR

This paper explores defining the Levy area of planar Brownian motion under Liouville measure for certain intermittency parameters, extending to smoother curves and analyzing properties of the resulting integration map.

Contribution

It introduces a novel approach to defining the Levy area in Gaussian multiplicative chaos for specific parameters and studies the properties of this new integration method.

Findings

Levy area can be defined for $oldsymbol{ ext{Brownian motion}}$ with Liouville measure when $oldsymbol{ ext{}\gamma< extstylerac{ ext{sqrt}(4/3)}{ ext{}}}$.

The paper extends the definition to smoother curves and investigates the properties of the associated integration map.

Provides new insights into stochastic integration within Gaussian multiplicative chaos framework.

Abstract

We show that for $\gamma<\sqrt{4/3}$, it is possible to define the Levy area of a planar Brownian motion with the Liouville measure of intermittency parameter $\gamma$ as the underlying area measure. We also consider the case of smoother curves, and study some properties of the integration map thus defined.