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Fisher information and trajectorial interpretation to the Itô--Langevin relative entropy dissipation

Jiaming Chen

TL;DR

The paper develops a trajectorial framework for the dissipation of relative entropy in Itô–Langevin dynamics by exploiting time-reversal of diffusions and embedding the evolution in Wasserstein space. By semimartingale decompositions of the backward-time likelihood ratio and its logarithm, the authors derive pathwise expressions for entropy dissipation that recover the classical de Bruijn identity and, via Wasserstein geometry, reveal the steepest-descent character of entropic evolution. The work clarifies perturbation effects through a gradient perturbation $\beta$ and shows how backward-time analysis yields tractable, interpretable formulas that connect entropy, Fisher information, and transport distance, including exponential decay rates under curvature conditions. Altogether, the results illustrate how trajectorial methods provide deeper insight into stochastic dissipative systems and their geometric structure, with implications for analysis of entropy production and gradient-flow dynamics in high-dimensional stochastic models.

Abstract

The dissipation phenomena of relative entropy from an Itô--Langevin dynamical system is a classic topic from stochastic analysis. Relying on the time-reversal of diffusions, a novel trajectorial approach investigates the pathwise behavior of relevant entropy processes, reveals more information from the delicate random structure, and eventually retrieves the known classical results. In essence, this approach provides novel insights and rederives the known results of the Itô--Langevin dynamics, as will be presented in this expository article. Another part is to view the stochastic time-evolution through the lens of the Wasserstein space, under which we observe the geometric feature of steepest descent of the entropy decay as well as its exponential rate of velocity.

Fisher information and trajectorial interpretation to the Itô--Langevin relative entropy dissipation

TL;DR

The paper develops a trajectorial framework for the dissipation of relative entropy in Itô–Langevin dynamics by exploiting time-reversal of diffusions and embedding the evolution in Wasserstein space. By semimartingale decompositions of the backward-time likelihood ratio and its logarithm, the authors derive pathwise expressions for entropy dissipation that recover the classical de Bruijn identity and, via Wasserstein geometry, reveal the steepest-descent character of entropic evolution. The work clarifies perturbation effects through a gradient perturbation and shows how backward-time analysis yields tractable, interpretable formulas that connect entropy, Fisher information, and transport distance, including exponential decay rates under curvature conditions. Altogether, the results illustrate how trajectorial methods provide deeper insight into stochastic dissipative systems and their geometric structure, with implications for analysis of entropy production and gradient-flow dynamics in high-dimensional stochastic models.

Abstract

The dissipation phenomena of relative entropy from an Itô--Langevin dynamical system is a classic topic from stochastic analysis. Relying on the time-reversal of diffusions, a novel trajectorial approach investigates the pathwise behavior of relevant entropy processes, reveals more information from the delicate random structure, and eventually retrieves the known classical results. In essence, this approach provides novel insights and rederives the known results of the Itô--Langevin dynamics, as will be presented in this expository article. Another part is to view the stochastic time-evolution through the lens of the Wasserstein space, under which we observe the geometric feature of steepest descent of the entropy decay as well as its exponential rate of velocity.

Paper Structure

This paper contains 27 sections, 24 theorems, 172 equations.

Table of Contents

  1. Introduction
  2. Reviewing the literature
  3. Motivation and rough descriptions
  4. Structure of this article
  5. The stochastic Itô--Langevin dynamics
  6. Itô--Langevin dynamics
  7. Density and reference measure
  8. Quantities from statistical physics
  9. Preview of classical results
  10. Time-reversal of diffusion processes
  11. Historical comments
  12. Backward filtrations
  13. Wiener process and Itô integration
  14. Diffusions under time-reversal
  15. Semimartingale decomposition of $\ell^\beta$ and $\mathcal{R}^\beta$
  16. ...and 12 more sections

Key Result

Lemma 2.1

The Itô-Langevin diffusion (perturbed Itô-Langevin dynamics) with initial distribution $P_0$ admits a pathwise unique strong solution $(X_t)_{t\geq0}$, whose distribution on $\mathcal{C}$ is denoted by $\mathbb{P}^\beta$. If we assume that the distribution $P_0$ of $X_0$ admits a finite second momen then each $X_t$, with $t\geq0$, admits a finite second moment, i.e. $\mathbb{E}^{\mathbb{P}^\beta}[

Theorems & Definitions (43)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 33 more