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

2024/03/26 by Herry, Ronan, Huguet, Baptiste
#49Q20 (Secondary) #60G53 (Primary) #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2403.17305

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.

Related

• Minimizing relative entropy of path measures under marginal constraints
• Benamou-Brenier and duality formulas for the entropic cost on RCD^*(K,N) spaces
• Small noise limit and convexity for generalized incompressible flows,\n Schr "odinger problems, and optimal transport
• Non-local Hamilton-Jacobi-Bellman equations for the stochastic optimal control of path-dependent piecewise deterministic processes
• A system of Schrödinger's problems and functional equations