Convergence of the dynamical discrete web to the dynamical Brownian web

TL;DR

This paper proves that the scaled dynamical discrete web converges to the dynamical Brownian web in path space topology, establishing a weak convergence result for these stochastic processes.

Contribution

It demonstrates the convergence of the dynamical discrete web to the dynamical Brownian web in the space of RCLL paths, providing a rigorous link between discrete and continuous models.

Findings

Almost sure RCLL paths for the DyBW

Weak convergence of scaled DyDW to DyBW

Path space topology convergence proof

Abstract

In this paper we study the convergence of dynamical discrete web (DyDW) to the dynamical Brownian web (DyBW) in the path space topology. We show that almost surely the DyBW has RCLL paths taking values in an appropriate metric space and as a sequence of RCLL paths, the scaled dynamical discrete web converges to the DyBW. This proves weak convergence of the DyDW process to the DyBW process.