Large deviations of the stochastic area for linear diffusions

Area of Science:

• Mathematical Physics
• Stochastic Processes
• Ergodic Theory

Background:

• The stochastic area of planar Brownian motion was studied by Lévy.
• For linear stochastic differential equations (SDEs), only the expected value of the stochastic area is known.

Purpose of the Study:

• To calculate the generating function of the stochastic area for linear SDEs.
• To extract large deviation functions and an effective SDE for long-time behavior.
• To obtain asymptotic mean and variance of the stochastic area.

Main Methods:

• Calculation of the generating function for stochastic area.
• Analysis of the generating function to derive large deviation functions.
• Derivation of asymptotic mean and variance from the generating function.

Main Results:

• A method to compute the generating function for stochastic area in linear SDEs.
• Identification of large deviation functions and an effective SDE for the long-time limit.
• Calculation of asymptotic mean and variance, crucial for diffusion reversibility analysis.

Conclusions:

• The study provides a comprehensive framework for analyzing the stochastic area in linear SDEs.
• Results offer insights into the long-time behavior and reversibility of diffusion processes.
• The developed methods can be applied to study both reversible and irreversible linear SDEs.