Skorokhod's M1 topology for distribution-valued processes

TL;DR

This paper extends Skorokhod's M1 topology to distribution-valued processes, providing tools for analyzing convergence of empirical processes in distributional evolution equations with boundary conditions.

Contribution

It introduces a new M1 topology for distribution-valued cadlag processes and develops compactness and tightness criteria for these processes.

Findings

Established M1 topology for distribution-valued processes.

Derived compactness and tightness characterizations.

Applied to convergence analysis of empirical process approximations.

Abstract

Skorokhod's M1 topology is defined for c\`adl\`ag paths taking values in the space of tempered distributions (more generally, in the dual of a countably Hilbertian nuclear space). Compactness and tightness characterisations are derived which allow us to study a collection of stochastic processes through their projections on the familiar space of real-valued c\`adl\`ag processes. It is shown how this topological space can be used in analysing the convergence of empirical process approximations to distribution-valued evolution equations with Dirichlet boundary conditions.