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Inversions of stochastic processes from ergodic measures of Nonlinear SDEs

Hongyu Liu, Zhihui Liu

TL;DR

This work formulates and analyzes the inverse problem of recovering drift and diffusion terms from an ergodic invariant measure of nonlinear SDEs and SPDEs, converting identifiability into a stationary Fokker–Planck equation framework. It establishes precise identifiability results in several key settings (notably 1D drift, Langevin-type models, and additive-noise SPDEs) while constructing counterexamples that highlight inherent non-uniqueness in other regimes. The authors provide explicit inverse representations for the coefficients in terms of invariant densities and extend the approach to infinite-dimensional SPDEs with Gibbs-type invariant measures. The results lay a rigorous theoretical foundation for equity-based recovery from equilibrium data and motivate development of practical algorithms for equilibrium-informed system identification.

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.

Inversions of stochastic processes from ergodic measures of Nonlinear SDEs

TL;DR

This work formulates and analyzes the inverse problem of recovering drift and diffusion terms from an ergodic invariant measure of nonlinear SDEs and SPDEs, converting identifiability into a stationary Fokker–Planck equation framework. It establishes precise identifiability results in several key settings (notably 1D drift, Langevin-type models, and additive-noise SPDEs) while constructing counterexamples that highlight inherent non-uniqueness in other regimes. The authors provide explicit inverse representations for the coefficients in terms of invariant densities and extend the approach to infinite-dimensional SPDEs with Gibbs-type invariant measures. The results lay a rigorous theoretical foundation for equity-based recovery from equilibrium data and motivate development of practical algorithms for equilibrium-informed system identification.

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.

Paper Structure

This paper contains 8 sections, 7 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fokker--Planck equation and Invariant measure
  4. Measurement operator
  5. Drift and Diffusion Inversion Problem for SDEs
  6. Drift inversion for SODEs
  7. Diffusion inversion for SODEs
  8. Drift and diffusion Inversion for SPDEs driven by additive noise

Key Result

Theorem 3.1

For a fixed $D=D_k \in \mathcal{C}(\mathbb R; \mathbb R)$, assume that $(b_k, D_k) \in \mathcal{A}$ and where $U_k \in \mathcal{C}^1(\mathbb R; \mathbb R)$ is any premitive of $b_k/D$, i.e., $U_k'=b_k /D$, $k=1,2$. Then $\pi_{b_1, D}=\pi_{b_2, D}$ if and only if $b_1=b_2$, i.e., $T_b$ is a bijective.

Theorems & Definitions (19)

  • Theorem 3.1
  • proof
  • Remark 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • proof
  • Theorem 3.5
  • proof
  • ...and 9 more