[Paper Review] Hopf-Cole transformation via generalized Schrödinger bridge problem

Flavien Léger, Wuchen Li|arXiv (Cornell University)|Jan 25, 2019

Nonlinear Waves and Solitons37 references4 citations

TL;DR

This paper establishes that generalized Hopf–Cole transformations, derived from the Schrödinger bridge problem, are symplectic submersions in Wasserstein symplectic geometry, providing a unified framework for fluid dynamics and optimal transport. The key contribution is a rigorous symplectic structure underlying these transformations, enabling energy splitting inequalities and extending to finite-dimensional graphs and manifolds.

ABSTRACT

We study generalized Hopf-Cole transformations motivated by the Schrödinger bridge problem, which can be seen as a boundary value Hamiltonian system on the Wasserstein space. We prove that generalized Hopf-Cole transformations are symplectic submersions in the Wasserstein symplectic geometry. Many examples, including a Hopf-Cole transformation for the shallow water equations, are given. Based on this transformation, energy splitting inequalities are provided.

Motivation & Objective

• To establish a symplectic geometric framework for generalized Hopf–Cole transformations arising from the Schrödinger bridge problem.
• To demonstrate that these transformations are symplectic submersions on the Wasserstein space of probability densities.
• To derive energy splitting inequalities using the symplectic structure of the transformed variables.
• To extend the formalism to finite-dimensional manifolds and graphs with homogeneous metrics.
• To connect the Hopf–Cole transformation to Hamiltonian flows in optimal transport and mean field games.

Proposed method

• Formulate the generalized Schrödinger bridge problem as a controlled gradient flow on the density manifold with general potential energy.
• Introduce the generalized Hopf–Cole transformation via the variable $ S_t = abla ilde{ ho}_t $, linking density $ \rho $ and potential $ \Phi $.
• Prove that the transformation is a symplectic submersion by showing preservation of the Wasserstein symplectic form.
• Derive the Hamiltonian system in transformed variables $ (\eta, \eta^*) $, where $ \eta_i = \sqrt{\rho_i} e^{S_i/2} $, $ \eta^*_i = \sqrt{\rho_i} e^{-S_i/2} $.
• Establish energy splitting inequalities using the symplectic structure, particularly for the discrete case on graphs.
• Verify the transformation’s consistency with the continuous limit, recovering nonlinear Laplacian reformulations.

Experimental results

Research questions

• RQ1 How can the Hopf–Cole transformation be generalized beyond the standard Burgers' equation to arbitrary potential energies in optimal transport?
• RQ2 What is the symplectic geometric structure of the generalized Hopf–Cole transformation in the Wasserstein space?
• RQ3 Can energy splitting inequalities be derived from the symplectic invariance of the transformed variables?
• RQ4 How does the Hopf–Cole transformation extend to finite-dimensional graphs with homogeneous metrics?
• RQ5 What is the continuous limit of the discrete symplectic system on graphs, and how does it relate to the nonlinear Laplacian?

Key findings

• The generalized Hopf–Cole transformation is a symplectic submersion in the Wasserstein symplectic geometry, preserving the canonical symplectic form.
• The transformation maps the Schrödinger bridge problem’s dynamics into a Hamiltonian system on the $ (\eta, \eta^*) $-variables with Hamiltonian $ \mathcal{K}(\eta, \eta^*) = -2\eta^T L(\eta\eta^*) \eta^* $.
• Energy splitting inequalities are derived by exploiting the symplectic invariance of the transformed system, providing functional bounds on the action.
• On graphs, the Hopf–Cole transformation leads to a symplectic Hamiltonian system with explicit continuity and Hamilton–Jacobi equations in $ (\eta, \eta^*) $.
• The continuous limit of the discrete system recovers a nonlinear Laplacian reformulation: $ -\Delta\eta = \frac{1}{2}\nabla \cdot (\eta\eta^* \nabla\eta) - \frac{1}{2}(\nabla\eta, \nabla\eta^*)\eta $.
• The formalism applies to the shallow water equations, revealing their underlying symplectic structure via the generalized Hopf–Cole transformation.