Stochastic Wasserstein Hamiltonian Flows

TL;DR

This paper explores stochastic Hamiltonian flows within the Wasserstein probability space, introducing a new variational approach that applies to quantum and stochastic systems, advancing understanding of stochastic dynamics in optimal transport.

Contribution

It introduces a novel variational formulation for stochastic Wasserstein Hamiltonian flows, extending their applicability to quantum and stochastic systems.

Findings

Stochastic Euler-Lagrange equations can be interpreted as stochastic kinetic Hamiltonian flows in Wasserstein space.

A new variational formulation for stochastic Wasserstein Hamiltonian flows is proposed.

The formulation applies to systems like stochastic Schrödinger equations and Schrödinger bridges.

Abstract

In this paper, we study the stochastic Hamiltonian flow in Wasserstein manifold, the probability density space equipped with $L^2$-Wasserstein metric tensor, via the Wong--Zakai approximation. We begin our investigation by showing that the stochastic Euler-Lagrange equation, regardless it is deduced from either variational principle or particle dynamics, can be interpreted as the stochastic kinetic Hamiltonian flows in Wasserstein manifold. We further propose a novel variational formulation to derive more general stochastic Wassersetin Hamiltonian flows, and demonstrate that this new formulation is applicable to various systems including the stochastic Schr\"odinger equation, Schr\"odinger equation with random dispersion, and Schr\"odinger bridge problem with common noise.