Inversions of stochastic processes from ergodic measures of Nonlinear SDEs

TL;DR

This paper studies the inverse problem of recovering drift and diffusion terms of nonlinear stochastic differential equations from their ergodic invariant measures, establishing conditions for unique identifiability and analyzing the underlying mathematical structure.

Contribution

It introduces a new theoretical framework for the unique recovery of stochastic process parameters from ergodic measures, extending classical inference methods to a broader class of systems.

Findings

Established conditions for unique identifiability of drift and diffusion from ergodic measures.

Analyzed differences between drift and diffusion inversion problems.

Provided counterexamples illustrating limitations of parameter recovery.

Abstract

We introduce and analyze a novel class of inverse problems for stochastic dynamics: Given the ergodic invariant measure of a stochastic process governed by a nonlinear stochastic ordinary or partial differential equation (SODE or SPDE), we investigate the unique identifiability of the underlying process--specifically, the recovery of its drift and diffusion terms. This stands in contrast to the classical problem of statistical inference from trajectory data. We establish unique identifiability results under several key scenarios, including cases with both multiplicative and additive noise, for both finite- and infinite-dimensional systems. Our analysis leverages the intrinsic structure of the governing equations and their quantitative relationship with the ergodic measure, thereby transforming the identifiability problem into a uniqueness issue for the solutions to the associated stationary Fokker-Planck equations. This approach reveals fundamental differences between drift and diffusion inversion problems and provides counterexamples where unique recovery fails. This work lays the theoretical foundation for a new research direction with significant potential for practical application.