We study systems of three interacting particles, in which drifts and variances are assigned by rank. These systems are degenerate: the variances corresponding to one or two ranks can vanish, so the corresponding ranked motions become ballistic rather than diffusive. Depending on which ranks are allowed to “go ballistic” the systems exhibit markedly different behavior, which we study here in some detail. Also studied are stability properties for the resulting planar process of gaps between successive ranks.
The first author was supported in part by the National Science Foundation under grants DMS-1615229 and DMS-2008427. The second author was supported by the National Science Foundation under grants NSF-DMS-0905754, NSF-DMS-1405210 and NSF-DMS-2004997.
We are grateful to Dr. E. Robert FERNHOLZ for initiating this line of research, prompting us over the years to continue it, and providing simulations for the paths of the processes involved. We are indebted to Drs. Jiro AKAHORI, David HOBSON, Chris ROGERS, Johannes RUF, Mykhaylo SHKOLNIKOV, Minghan YAN, and especially Andrey SARANTSEV, for discussions about the problems treated here, and for their advice and suggestions. The referee read the paper with great diligence and care, corrected a serious error, and made several incisive comments and suggestions, for which we are deeply grateful.
"Degenerate competing three-particle systems." Bernoulli 28 (3) 2067 - 2094, August 2022. https://doi.org/10.3150/21-BEJ1411