## The Annals of Probability

- Ann. Probab.
- Volume 15, Number 2 (1987), 659-675.

### Joint Continuity of the Intersection Local Times of Markov Processes

#### Abstract

We describe simple conditions on the transition density functions of two independent Markov processes $X$ and $Y$ which guarantee the existence of a continuous version for the intersection local time, formally given by $\alpha (z, H) = \int_H\int \delta_z (Y_t - X_s) ds dt$. In the analogous case of self-intersections $\alpha$ can be discontinuous at $z = 0$. We develop a Tanaka-like formula for $\alpha$ and use this to show that the singular part of $\alpha (z,\lbrack 0, T\rbrack^2)$ as $z \rightarrow 0$ is given by $2\int^T_0 U(X_t - z, X_t) dt, a.s.$, where $U$ is the 1-potential of $X$.

#### Article information

**Source**

Ann. Probab., Volume 15, Number 2 (1987), 659-675.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176992164

**Digital Object Identifier**

doi:10.1214/aop/1176992164

**Mathematical Reviews number (MathSciNet)**

MR885136

**Zentralblatt MATH identifier**

0622.60084

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60J25: Continuous-time Markov processes on general state spaces

Secondary: 60J55: Local time and additive functionals 60J60: Diffusion processes [See also 58J65]

**Keywords**

Markov processes intersection local time renormalization

#### Citation

Rosen, Jay. Joint Continuity of the Intersection Local Times of Markov Processes. Ann. Probab. 15 (1987), no. 2, 659--675. doi:10.1214/aop/1176992164. https://projecteuclid.org/euclid.aop/1176992164