## Electronic Communications in Probability

- Electron. Commun. Probab.
- Volume 21 (2016), paper no. 16, 12 pp.

### Discrete approximations to local times for reflected diffusions

#### Abstract

We propose a discrete analogue for the boundary local time of reflected diffusions in bounded Lipschitz domains. This discrete analogue, called the discrete local time, can be effectively simulated in practice and is obtained pathwise from random walks on lattices. We establish weak convergence of the joint law of the discrete local time and the associated random walks as the lattice size decreases to zero. A cornerstone of the proof is the local central limit theorem for reflected diffusions developed in [7]. Applications of the join convergence result to PDE problems are illustrated.

#### Article information

**Source**

Electron. Commun. Probab. Volume 21 (2016), paper no. 16, 12 pp.

**Dates**

Received: 10 November 2015

Accepted: 19 February 2016

First available in Project Euclid: 23 February 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ecp/1456238572

**Digital Object Identifier**

doi:10.1214/16-ECP4694

**Mathematical Reviews number (MathSciNet)**

MR3485385

**Zentralblatt MATH identifier**

1336.60068

**Subjects**

Primary: 60F17: Functional limit theorems; invariance principles 60J55: Local time and additive functionals

Secondary: 35K10: Second-order parabolic equations 35J25: Boundary value problems for second-order elliptic equations 49M25: Discrete approximations

**Keywords**

random walk reflected diffusion local time heat kernel Robin boundary problem

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Fan, Wai-Tong Louis. Discrete approximations to local times for reflected diffusions. Electron. Commun. Probab. 21 (2016), paper no. 16, 12 pp. doi:10.1214/16-ECP4694. https://projecteuclid.org/euclid.ecp/1456238572