Modified Massive Arratia flow and Wasserstein diffusion

TL;DR

This paper introduces a modified Arratia flow where particles carry mass affecting their diffusivity, relating it to Wasserstein diffusion and establishing a Varadhan formula for short-time behavior.

Contribution

It presents a new variant of Arratia flow with mass-dependent diffusivity and connects it to Wasserstein diffusion, providing a martingale solution and a short-time Varadhan formula.

Findings

The process is formulated as a martingale solution to a related SPDE.

A Varadhan formula for short times is established, governed by quadratic Wasserstein distance.

The model extends previous Arratia flow work by incorporating mass and diffusivity scaling.

Abstract

Extending previous work [arXiv:1408.0628] by the first author we present a variant of the Arratia flow, which consists of a collection of coalescing Brownian motions starting from every point of the unit interval. The important new feature of the model is that individual particles carry mass which aggregates upon coalescence and which scales the diffusivity of each particle in an inverse proportional way. In this work we relate the induced measure valued process to the Wasserstein diffusion of [arXiv:0704.0704]. First, we present the process as a martingale solution to a SPDE similar to [arXiv:0704.0704]. Second, as our main result we show a Varadhan formula for short times which is governed by the quadratic Wasserstein distance.