Equilibrium Moment Analysis of It\^o SDEs

TL;DR

This paper introduces a novel equilibrium-analysis method for Itô SDEs that leverages finite-timestep simulation logic to determine equilibrium moments, especially when the steady-state distribution's integral equation diverges.

Contribution

The paper presents a new technique for analyzing equilibrium moments of Itô SDEs that works even when traditional integral equations fail to converge.

Findings

Derived a relationship between raw moments of equilibrium distributions.

Applied the method to systems with non-convergent integral equations.

Provided insights into equilibrium properties of complex stochastic systems.

Abstract

Stochastic differential equations have proved to be a valuable governing framework for many real-world systems which exhibit ``noise'' or randomness in their evolution. One quality of interest in such systems is the shape of their equilibrium probability distribution, if such a thing exists. In some cases a straightforward integral equation may yield this steady-state distribution, but in other cases the equilibrium distribution exists and yet that integral equation diverges. Here we establish a new equilibrium-analysis technique based on the logic of finite-timestep simulation which allows us to glean information about the equilibrium regardless -- in particular, a relationship between the raw moments of the equilibrium distribution. We utilize this technique to extract information about one such equilibrium resistant to direct definition.