## The Annals of Probability

- Ann. Probab.
- Volume 37, Number 1 (2009), 78-106.

### Limits of one-dimensional diffusions

#### Abstract

In this paper, we look at the properties of limits of a sequence of real valued inhomogeneous diffusions. When convergence is only in the sense of finite-dimensional distributions, then the limit does not have to be a diffusion. However, we show that as long as the drift terms satisfy a Lipschitz condition and the limit is continuous in probability, then it will lie in a class of processes that we refer to as the *almost-continuous diffusions*. These processes are strong Markov and satisfy an “almost-continuity” condition. We also give a simple condition for the limit to be a continuous diffusion.

These results contrast with the multidimensional case where, as we show with an example, a sequence of two-dimensional martingale diffusions can converge to a process that is both discontinuous and non-Markov.

#### Article information

**Source**

Ann. Probab., Volume 37, Number 1 (2009), 78-106.

**Dates**

First available in Project Euclid: 17 February 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1234881685

**Digital Object Identifier**

doi:10.1214/08-AOP397

**Mathematical Reviews number (MathSciNet)**

MR2489160

**Zentralblatt MATH identifier**

1210.60087

**Subjects**

Primary: 60J60: Diffusion processes [See also 58J65] 60G44: Martingales with continuous parameter

Secondary: 60F99: None of the above, but in this section

**Keywords**

Diffusion martingale strong Markov finite-dimensional distributions

#### Citation

Lowther, George. Limits of one-dimensional diffusions. Ann. Probab. 37 (2009), no. 1, 78--106. doi:10.1214/08-AOP397. https://projecteuclid.org/euclid.aop/1234881685