## The Annals of Probability

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

Jay Rosen

#### 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

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

#### 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