Time-Reversal of Coalescing Diffusive Flows and Weak Convergence of Localized Disturbance Flows

TL;DR

This paper extends the coalescing Brownian web to include variable drift and diffusivity, identifies its time-reversal, and demonstrates weak convergence of localized disturbance flows to these coalescing diffusive flows.

Contribution

It introduces coalescing diffusive flows with variable parameters and characterizes their time-reversal, providing two proofs and convergence results for disturbance flows.

Findings

Time-reversal of coalescing diffusive flows is explicitly identified.

Localized disturbance flows converge weakly to coalescing diffusive flows.

Two distinct proofs for the time-reversal characterization are provided.

Abstract

We generalize the coalescing Brownian flow, aka the Brownian web, considered as a weak flow to allow varying drift and diffusivity in the constituent diffusion processes and call these flows coalescing diffusive flows. We then identify the time-reversal of each coalescing diffusive flow and provide two distinct proofs of this identification. One of which is direct and the other proceeds by generalizing the concept of a localized disturbance flow to allow varying size and shape of disturbances, we show these new flows converge weakly under appropriate conditions to a coalescing diffusive flow and identify their time-reversals.