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The Stefan problem and free targets of optimal Brownian martingale transport

Inwon C. Kim, Young-Heon Kim

TL;DR

This work introduces a free-target optimal Brownian stopping problem with a density upper bound f and derives a universal optimal target ν^* that is independent of the specific monotone cost type. By linking the stochastic barrier structure to a space–time Eulerian flow, the authors connect this probabilistic framework to the Stefan problem, obtaining global-time existence and weak–strong uniqueness results for the ill-posed St_1 and unifying it with the stable St_2 via a common barrier-driven mechanism. Central contributions include the universality and shadow interpretation of ν^*, the saturation property that ν = f on active regions, and a detailed Eulerian/parabolic-Obstacle perspective that yields monotonicity, L^1 contraction, BV estimates, and barrier-closure results. The results provide a variationally grounded, robust path from constrained stochastic transport to free-boundary PDEs, with implications for well-posedness, regularity, and initial-domain design in Stefan-type problems.

Abstract

We formulate and solve a free target optimal Brownian stopping problem from a given distribution while the target distribution is free and is conditioned to satisfy a given density height constraint. The free target optimization problem exhibits monotonicity, from which a remarkable universality follows, in the sense that the optimal target is independent of its Lagrangian cost type. In particular, the solutions to this optimization problem generate solutions to both unstable and stable type of the Stefan problem, where former stands for freezing of supercooled fluid $(St_1)$ and the latter for ice melting $(St_2)$. This unified approach to both types of Stefan problem is new. In particular we obtain global-time existence and weak-strong uniqueness for the ill-posed freezing problem $(St_1)$, for a given initial data and for a well-prepared class of initial domains generated from the initial data.

The Stefan problem and free targets of optimal Brownian martingale transport

TL;DR

This work introduces a free-target optimal Brownian stopping problem with a density upper bound f and derives a universal optimal target ν^* that is independent of the specific monotone cost type. By linking the stochastic barrier structure to a space–time Eulerian flow, the authors connect this probabilistic framework to the Stefan problem, obtaining global-time existence and weak–strong uniqueness results for the ill-posed St_1 and unifying it with the stable St_2 via a common barrier-driven mechanism. Central contributions include the universality and shadow interpretation of ν^*, the saturation property that ν = f on active regions, and a detailed Eulerian/parabolic-Obstacle perspective that yields monotonicity, L^1 contraction, BV estimates, and barrier-closure results. The results provide a variationally grounded, robust path from constrained stochastic transport to free-boundary PDEs, with implications for well-posedness, regularity, and initial-domain design in Stefan-type problems.

Abstract

We formulate and solve a free target optimal Brownian stopping problem from a given distribution while the target distribution is free and is conditioned to satisfy a given density height constraint. The free target optimization problem exhibits monotonicity, from which a remarkable universality follows, in the sense that the optimal target is independent of its Lagrangian cost type. In particular, the solutions to this optimization problem generate solutions to both unstable and stable type of the Stefan problem, where former stands for freezing of supercooled fluid and the latter for ice melting . This unified approach to both types of Stefan problem is new. In particular we obtain global-time existence and weak-strong uniqueness for the ill-posed freezing problem , for a given initial data and for a well-prepared class of initial domains generated from the initial data.

Paper Structure

This paper contains 30 sections, 44 theorems, 214 equations.

Table of Contents

  1. Introduction
  2. The connection between the Stefan problem, Problem \ref{['eqn:problem-OSP']}, and Problem \ref{['prob:mainquestion']}
  3. Remarks on future work
  4. Preliminaries
  5. Notation
  6. Randomized stopping times and optimal Skorokhod problem in $\mathbb{R}^d$
  7. Stopping times and randomized stopping times
  8. Subharmonic order and optimal Skorokhod problem
  9. Eulerian formulation
  10. Properties of stopping times
  11. Properties of stopping times with respect to a barrier set
  12. Optimal stopping times and potential flows
  13. Potential flows, definition and regularity
  14. Hitting times to monotone barriers and potential functions
  15. Remarks on the potential approach and the parabolic obstacle problem
  16. ...and 15 more sections

Key Result

Theorem 2.1

Given compactly supported measures $\mu, \nu$ on $\mathbb{R}^d$ with $\mu\le_{SH}\nu$ and $\mu, \nu \ll Leb$, the optimal Skorokhod problem $\mathcal{P}(\mu, \nu)$eqn:problem-OSP with the cost $\mathcal{C}$ in eqn:Lag-cost under the assumption eqn:boundedness-assumption and eqn:monotone-assumption,

Theorems & Definitions (109)

  • Theorem 2.1: Existence/Uniqueness of Optimal Skorokhod Problem beiglboeck2017optimal PDE19
  • Proposition 2.2: see PDE19 Proposition 2.2
  • Lemma 2.3: PDE19 Lemma 4.5
  • Remark 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Remark 3.3
  • proof
  • Corollary 3.4
  • ...and 99 more