This paper is concerned with the intrinsic metrics of the two main classes of superprocesses. For the Fleming-Viot process, we identify it as the Bhattacharya distance, and for Dawson-Watanabe processes, we find the Kakutani-Hellinger metric. The corresponding geometries are studied in some detail. In particular, representation formulas for geodesics and arc length functionals are obtained. The relations between the two metrics yield a geometric interpretation of the identification of the Fleming-Viot process as a Dawson-Watanabe superprocess conditioned to have total mass 1. As an application, a functional limit theorem for super-Brownian motion conditioned on local extinction is proved.
"Geometric aspects of Fleming-Viot and Dawson-Watanabe processes." Ann. Probab. 25 (3) 1160 - 1179, July 1997. https://doi.org/10.1214/aop/1024404509