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The Brenier-Schrödinger problem with respect to Feller semimartingales and non-local Hamilton-Jacobi-Bellman equations

Ronan Herry, Baptiste Huguet

Abstract

Motivated by a problem from incompressible fluid mechanics of Brenier (JAMS 1989), and its recent entropic relaxation by Arnaudo, Cruizero, Léonard & Zambrini (AIHP PS 2020), we study a problem of entropic minimization on the path space when the reference measure is a generic Feller semimartingale. We show that, under some regularity condition, our problem connects naturally with a, possibly non-local, version of the Hamilton-Jacobi-Bellman equation. Additionally, we study existence of minimizers when the reference measure in a Ornstein-Uhlenbeck process.

The Brenier-Schrödinger problem with respect to Feller semimartingales and non-local Hamilton-Jacobi-Bellman equations

Abstract

Motivated by a problem from incompressible fluid mechanics of Brenier (JAMS 1989), and its recent entropic relaxation by Arnaudo, Cruizero, Léonard & Zambrini (AIHP PS 2020), we study a problem of entropic minimization on the path space when the reference measure is a generic Feller semimartingale. We show that, under some regularity condition, our problem connects naturally with a, possibly non-local, version of the Hamilton-Jacobi-Bellman equation. Additionally, we study existence of minimizers when the reference measure in a Ornstein-Uhlenbeck process.
Paper Structure (47 sections, 10 theorems, 87 equations)

This paper contains 47 sections, 10 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Motivations and connections to the literature
  4. Outline of the proof
  5. Reminders and notation
  6. Semimartingales and their characteristics
  7. Path space
  8. Processes
  9. Martingales & semimartingales
  10. Special semimartingales
  11. Canonical jumps measure
  12. Compensator of the jumps measure
  13. Characteristics of a semimartingale
  14. Representation of semimartingales with given characteristics
  15. Markov processes and related concepts
  16. ...and 32 more sections

Key Result

Theorem 1

Consider a regular solution $\mathsf{P}$ of the aforementioned problem, and the associated $\psi^{x}$. Then, the additive functional $A$ is absolutely continuous. In particular, there exists a function $p \colon [0,1] \times \mathbb{R}^{n} \to \mathbb{R}$ such that Moreover, $\psi^{x}$ is a solution to the generalized Hamilton--Jacobi--Bellman equation where $\mathbf{A}$ denotes the Markov gener

Theorems & Definitions (23)

  • Theorem
  • Remark 1.1
  • Remark 2.1
  • Theorem 2.2: Leonard
  • Remark 2.3
  • Proposition 2.4: Leonard
  • Remark 2.5
  • Remark 2.6
  • Theorem 3.1
  • Lemma 3.2
  • ...and 13 more