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April, 1987 Joint Continuity of the Intersection Local Times of Markov Processes
Jay Rosen
Ann. Probab. 15(2): 659-675 (April, 1987). DOI: 10.1214/aop/1176992164

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 $X$.

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Jay Rosen. "Joint Continuity of the Intersection Local Times of Markov Processes." Ann. Probab. 15 (2) 659 - 675, April, 1987. https://doi.org/10.1214/aop/1176992164

Information

Published: April, 1987
First available in Project Euclid: 19 April 2007

zbMATH: 0622.60084
MathSciNet: MR885136
Digital Object Identifier: 10.1214/aop/1176992164

Subjects:
Primary: 60J25
Secondary: 60J55 , 60J60

Keywords: Intersection local time , Markov processes , renormalization

Rights: Copyright © 1987 Institute of Mathematical Statistics

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Vol.15 • No. 2 • April, 1987
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