This paper examines the existence of the self-intersection local time for a superprocess over a stochastic flow in dimensions $d\leq3$, which through constructive methods, results in a Tanaka-like representation. The superprocess over a stochastic flow is a superprocess with dependent spatial motion, and thus Dynkin’s proof of existence, which requires multiplicity of the log-Laplace functional, no longer applies. Skoulakis and Adler’s method of calculating moments is extended to higher moments, from which existence follows.
"Generalized self-intersection local time for a superprocess over a stochastic flow." Ann. Probab. 40 (4) 1483 - 1534, July 2012. https://doi.org/10.1214/11-AOP653