## The Annals of Probability

### On One-Dimensional Diffusions with Time Parameter Set $(-\infty, \infty)$

J. Theodore Cox

#### Abstract

Let $p_t, t \geqq 0$ be the probability transition semigroup for a continuous one-dimensional diffusion. We examine continuous Markov processes $\xi_s$, defined for all $-\infty < s < \infty$, which are governed by $p_t$. We determine necessary and sufficient conditions for the set of such processes governed by $p_t$ to be nontrivial, and give an example where these conditions are satisfied.

#### Article information

Source
Ann. Probab., Volume 5, Number 5 (1977), 807-813.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176995724

Digital Object Identifier
doi:10.1214/aop/1176995724

Mathematical Reviews number (MathSciNet)
MR448587

Zentralblatt MATH identifier
0376.60078

JSTOR
Cox, J. Theodore. On One-Dimensional Diffusions with Time Parameter Set $(-\infty, \infty)$. Ann. Probab. 5 (1977), no. 5, 807--813. doi:10.1214/aop/1176995724. https://projecteuclid.org/euclid.aop/1176995724