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Stochastic optimal transport for the Langevin dynamics and its zero--mass limit

Toshio Mikami

TL;DR

The paper develops a stochastic optimal transport framework for Langevin dynamics with positive mass $m>0$, fixing only the initial and terminal position distributions while allowing uncertainty in initial momentum. It proves existence of minimizers, establishes compactness and convergence in the zero-mass limit, and shows that minimizers converge to those of the standard SOT for continuous semimartingales, with the limit enforcing $Y(0)\to0$ and $P^{(X(0),X(1))}\in \Pi(P_0,P_1)$. A dual formulation is provided to elucidate the structure of the problem and explain the necessity of considering a family of SOTs indexed by the initial momentum set. The results hold for general cost functions and extend to polynomial-growth costs, including a Schrödinger-type duality scenario for $L=|u|^2/2$. Collectively, the work links finite-inertia Langevin dynamics to the overdamped zero-mass regime within a rigorous variational and probabilistic framework, with implications for information theory and data-driven transport that accounts for velocity uncertainty.

Abstract

We introduce a stochastic optimal transport for the Langevin dynamics with positive mass and study its zero--mass limit. The new aspect of this paper is that we only fix the initial and terminal probability distributions of the positions of particles under consideration, but not those of their velocities with Heisenberg's uncertainty principle in mind. In the zero--mass limit, we show that the minimizer of our stochastic optimal transport is tight if and only if the initial momentum of a particle converges to zero. We also show that the limit of a minimizer of our stochastic optimal transport is a minimizer of a standard stochastic optimal transport for continuous semimartingales.

Stochastic optimal transport for the Langevin dynamics and its zero--mass limit

TL;DR

The paper develops a stochastic optimal transport framework for Langevin dynamics with positive mass , fixing only the initial and terminal position distributions while allowing uncertainty in initial momentum. It proves existence of minimizers, establishes compactness and convergence in the zero-mass limit, and shows that minimizers converge to those of the standard SOT for continuous semimartingales, with the limit enforcing and . A dual formulation is provided to elucidate the structure of the problem and explain the necessity of considering a family of SOTs indexed by the initial momentum set. The results hold for general cost functions and extend to polynomial-growth costs, including a Schrödinger-type duality scenario for . Collectively, the work links finite-inertia Langevin dynamics to the overdamped zero-mass regime within a rigorous variational and probabilistic framework, with implications for information theory and data-driven transport that accounts for velocity uncertainty.

Abstract

We introduce a stochastic optimal transport for the Langevin dynamics with positive mass and study its zero--mass limit. The new aspect of this paper is that we only fix the initial and terminal probability distributions of the positions of particles under consideration, but not those of their velocities with Heisenberg's uncertainty principle in mind. In the zero--mass limit, we show that the minimizer of our stochastic optimal transport is tight if and only if the initial momentum of a particle converges to zero. We also show that the limit of a minimizer of our stochastic optimal transport is a minimizer of a standard stochastic optimal transport for continuous semimartingales.
Paper Structure (7 sections, 18 theorems, 222 equations)

This paper contains 7 sections, 18 theorems, 222 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Zero--mass limit: general cost function
  4. Zero--mass limit: cost function of polynomial growth in $u$
  5. Duality formula
  6. Lemmas
  7. Proofs of main results

Key Result

Theorem 2.1

Suppose that (A1, i, ii) holds and that $m>0$. Then for any closed set $B\subset \mathcal{P}(\mathbb{R}^d)$, any $P_0,P_1\in\mathcal{P}(\mathbb{R}^d)$, and for any $\{Z_n=(X_n, Y_n)\}_{n\ge 1}\subset \mathcal{A}^m(B,P_0;P_1)$ such that and $\{Z_n\}_{n\ge 1}$ are tight. For any weak limit point $Q_\infty(dtdzdu)$ of $\{Q_n(dtdzdu)\}_{n\ge 1}$, there exists $Z=(X,Y)\in \mathcal{A}^m(B,P_0;P_1)$ suc

Theorems & Definitions (40)

  • Remark 1.1
  • Remark 1.2
  • Definition 1.1: SOT for the Langevin dynamics with positive mass
  • Remark 1.3
  • Definition 1.2
  • Remark 1.4
  • Remark 2.1
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • ...and 30 more