Fluctuations in Wasserstein dynamics on Graphs

TL;DR

This paper introduces a novel stochastic diffusion process on the probability simplex, derived from chemical reaction models, with applications to Wasserstein dynamics on graphs and connections to Wright-Fisher diffusion.

Contribution

It develops a diffusion approximation for chemical reaction counting processes, linking gradient flows, Riemannian metrics, and Langevin dynamics on probability simplices.

Findings

Derivation of Langevin dynamics with degenerate Brownian motion on the probability simplex.

Connection of the diffusion process to Wright-Fisher diffusion in two-point cases.

Identification of the invariant Gibbs measure as the stationary distribution.

Abstract

In this paper, we propose a drift-diffusion process on the probability simplex to study stochastic fluctuations in probability spaces. We construct a counting process for linear detailed balanced chemical reactions with finite species such that its thermodynamic limit is a system of ordinary differential equations (ODE) on the probability simplex. This ODE can be formulated as a gradient flow with an Onsager response matrix that induces a Riemannian metric on the probability simplex. After incorporating the induced Riemannian structure, we propose a diffusion approximation of the rescaled counting process for molecular species in the chemical reactions, which leads to Langevin dynamics on the probability simplex with a degenerate Brownian motion constructed from the eigen-decomposition of Onsager's response matrix. The corresponding Fokker-Planck equation on the simplex can be regarded as the usual drift-diffusion equation with the extrinsic representation of the divergence and Laplace-Beltrami operator. The invariant measure is the Gibbs measure, which is the product of the original macroscopic free energy and a volume element. When the drift term vanishes, the Fokker-Planck equation reduces to the heat equation with the Laplace-Beltrami operator, which we refer to as canonical Wasserstein diffusions on graphs. In the case of a two-point probability simplex, the constructed diffusion process is converted to one dimensional Wright-Fisher diffusion process, which leads to a natural boundary condition ensuring that the process remains within the probability simplex.