Equilibrium Moment Analysis of It\^o SDEs
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.
Cite
@article{arxiv.2502.00918,
title = {Equilibrium Moment Analysis of It\^o SDEs},
author = {David Sabin-Miller and Daniel M. Abrams},
journal= {arXiv preprint arXiv:2502.00918},
year = {2025}
}
Comments
Main text: 4 pages, 1 figure. Supplemental Material: 3 pages, 4 figures. arXiv admin note: substantial text overlap with arXiv:2210.03781