Equilibrium Moment Analysis of Itô SDEs

TL;DR

The paper addresses the challenge of characterizing equilibrium distributions of Itô SDEs when direct analytical forms are intractable or diverge. It introduces a finite-timestep, Euler–Maruyama–guided moment-consistency framework that yields explicit relations among equilibrium moments (starting with $\mu_2$ and generalizing to higher moments) at leading order in $\Delta t$. Applying the method to a cubic stochastic attractor reveals a critical noise threshold $\sigma > \sqrt{2}/3$ beyond which finite moments cannot satisfy the consistency relations, indicating moment divergence; higher-moment analysis provides recursive constraints and, with a near-Gaussian assumption, matches numerical simulations in the finite-moment regime. The approach offers a practical tool for inferring equilibrium structure and detecting divergence regimes in systems where the equilibrium PDF is not readily obtainable, with potential to fully specify equilibrium moments in favorable cases.

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.

Figures (1)

• Figure 1: Numerical validation. Comparison of numerical results (via Fokker-Planck evolution) to the theoretical relation, augmented with the extra Gaussian condition $\mu_4 = 3 \mu_2^2$. The shaded region indicates $\mu_2 < 0.1 \sigma^2$ (top) and correspondingly $\mu_4 < 0.03 \sigma^4$ (bottom), where the Gaussian approximation (from $\mu_2 \ll \sigma^2$) should be most valid. Top: Smaller $\sigma$ values take longer simulated time $T$ to equilibrate, but do approach the theorized line in the shaded region. For high noise amplitudes, the relation need not hold, and indeed theory suggests that $\mu_2$ and $\mu_4$ should diverge for $\sigma>\sqrt{2}/3$ (indicated by the vertical dashed line)---though this divergence is invisible at constant domain width. Bottom: As predicted by theory, the fourth moment $\mu_4$ does indeed appear to diverge for $\sigma>\sqrt{2}/3$, though simulation with ever wider domain width $W$ (measured in number of standard deviations of the equilibrium solution) is needed capture more of the distribution's tails (all curves shown for $T = 100$).