Large deviations of the stochastic area for linear diffusions

TL;DR

This paper derives the large deviation properties of the stochastic area for linear diffusions, providing new analytical tools to understand the probability of rare events and the reversibility of diffusive processes.

Contribution

It introduces a method to compute the generating function and large deviation functions for the stochastic area in linear SDEs, extending classical results to more general diffusions.

Findings

Derived the generating function for stochastic area in linear SDEs

Obtained large deviation functions characterizing rare events

Analyzed examples of reversible and irreversible diffusions

Abstract

The area enclosed by the two-dimensional Brownian motion in the plane was studied by L\'evy, who found the characteristic function and probability density of this random variable. For other planar processes, in particular ergodic diffusions described by linear stochastic differential equations (SDEs), only the expected value of the stochastic area is known. Here, we calculate the generating function of the stochastic area for linear SDEs, which can be related to the integral of the angular momentum, and extract from the result the large deviation functions characterising the dominant part of its probability density in the long-time limit, as well as the effective SDE describing how large deviations arise in that limit. In addition, we obtain the asymptotic mean of the stochastic area, which is known to be related to the probability current, and the asymptotic variance, which is important for determining from observed trajectories whether or not a diffusion is reversible. Examples of reversible and irreversible linear SDEs are studied to illustrate our results.