[Paper Review] A Hamilton-Jacobi PDE associated with hydrodynamic fluctuations from a nonlinear diffusion equation

Jin Feng, Toshio Mikami+1 more|arXiv (Cornell University)|Feb 28, 2019

Stochastic processes and financial applicationsEconomics, Econometrics and Finance29 references3 citations

TL;DR

This paper establishes a Hamilton-Jacobi PDE in the space of probability measures arising from hydrodynamic fluctuations in a nonlinear diffusion process, using a two-scale averaging method on a stochastic particle model (Carleman-type). It proves uniqueness via comparison principles, constructs solutions via controlled PDEs, and shows the Hamiltonian emerges naturally from microscopic dynamics, offering a new scale-bridging framework for stochastic hydrodynamics with potential extension to deterministic systems.

ABSTRACT

We study a class of Hamilton-Jacobi partial differential equations in the space of probability measures. In the first part of this paper, we prove comparison principles (implying uniqueness) for this class. In the second part, we establish the existence of a solution and give a representation using a family of partial differential equations with control. A large part of our analysis exploits special structures of the Hamiltonian, which might look mysterious at first sight. However, we show that this Hamiltonian structure arises naturally as limit of Hamiltonians of microscopical models. Indeed, in the third part of this paper, we informally derive the Hamiltonian studied before, in a context of fluctuation theory on the hydrodynamic scale. The analysis is carried out for a specific model of stochastic interacting particles in gas kinetics, namely a version of the Carleman model. We use a two-scale averaging method on Hamiltonians defined in the space of probability measures to derive the limiting Hamiltonian.

Motivation & Objective

• To develop a variational, large deviation-based approach to derive macroscopic hydrodynamic equations from stochastic particle systems, avoiding reliance on traditional probabilistic ergodic theory.
• To establish well-posedness of a novel class of Hamilton-Jacobi equations in the space of probability measures, focusing on uniqueness and existence of solutions.
• To derive the limiting Hamiltonian structure from microscopic stochastic dynamics using a two-scale averaging method in the space of probability measures.
• To connect the resulting Hamiltonian to weak KAM theory and infinite-particle systems, enabling a bridge between microscopic dynamics and macroscopic fluctuations.
• To lay a foundation for extending such methods to deterministic hydrodynamic limits by formalizing a scale-bridging mechanism rooted in Hamiltonian structure.

Proposed method

• Prove comparison principles for Hamilton-Jacobi PDEs in the space of probability measures, ensuring uniqueness of viscosity solutions.
• Construct solutions via a family of controlled PDEs, representing the value function as an infimum over control processes.
• Apply a two-scale averaging method to Hamiltonians defined on probability measures, extracting the effective macroscopic Hamiltonian from fast-slow particle dynamics.
• Use the Carleman model of stochastic interacting particles as the microscopic model to derive the limiting Hamiltonian structure.
• Leverage weak KAM theory and functional-analytic techniques to analyze the Hamiltonian in the space of probability measures.
• Establish equivalence between Wasserstein-1 and negative Sobolev norms to relate metric structures in the space of measures.

Experimental results

Research questions

• RQ1 How can a Hamilton-Jacobi PDE in the space of probability measures be rigorously formulated and solved for hydrodynamic fluctuations?
• RQ2 What is the role of the Hamiltonian structure in the emergence of macroscopic behavior from stochastic particle systems?
• RQ3 Can a two-scale averaging method in the space of probability measures yield a well-defined limiting Hamiltonian for fluctuation theory?
• RQ4 How does the derived Hamiltonian relate to weak KAM theory and infinite-particle systems?
• RQ5 To what extent can this framework be generalized to deterministic hydrodynamic limits?

Key findings

• Comparison principles are established for the class of Hamilton-Jacobi PDEs in the space of probability measures, guaranteeing uniqueness of viscosity solutions.
• A solution to the limiting Hamilton-Jacobi equation is constructed using a representation via controlled PDEs, confirming existence.
• The limiting Hamiltonian arises naturally as the two-scale limit of microscopic Hamiltonians from a stochastic particle model (Carleman model).
• The method provides a functional-analytic alternative to the block-averaging and replacement-lemma techniques in large deviation theory.
• The analysis reveals a deep connection between the Hamiltonian structure and weak KAM theory in infinite-dimensional measure spaces.
• A quantitative estimate is derived: $\|\rho - \gamma\|_{-1}^2 \leq \frac{4}{\pi} W_1(\rho, \gamma)$, linking negative Sobolev and Wasserstein-1 norms.