## Electronic Communications in Probability

- Electron. Commun. Probab.
- Volume 21 (2016), paper no. 34, 11 pp.

### Skorokhod’s M1 topology for distribution-valued processes

#### Abstract

Skorokhod’s M1 topology is defined for càdlàg 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 càdlàg 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.

#### Article information

**Source**

Electron. Commun. Probab., Volume 21 (2016), paper no. 34, 11 pp.

**Dates**

Received: 10 December 2015

Accepted: 11 April 2016

First available in Project Euclid: 21 April 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ecp/1461251162

**Digital Object Identifier**

doi:10.1214/16-ECP4754

**Mathematical Reviews number (MathSciNet)**

MR3492929

**Zentralblatt MATH identifier**

1338.60105

**Subjects**

Primary: 60G07: General theory of processes 60F17: Functional limit theorems; invariance principles

**Keywords**

Skorokhod M1 topology compacntess and tightness characterisation tempered distribution countably Hilbertian nuclear space

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Ledger, Sean. Skorokhod’s M1 topology for distribution-valued processes. Electron. Commun. Probab. 21 (2016), paper no. 34, 11 pp. doi:10.1214/16-ECP4754. https://projecteuclid.org/euclid.ecp/1461251162