@article{c3a4b65df4f34527a05acd1269f1e40d,

title = "Weakly mixing smooth planar vector field without asymptotic directions",

abstract = "We construct a planar smooth weakly mixing stationary random vector field with nonnegative components such that, with probability 1, the flow generated by this vector field does not have an asymptotic direction. Moreover, for all individual trajectories, the set of partial limiting directions coincides with those spanning the positive quadrant. A modified example shows that a particle in space-time weakly mixing positive velocity field does not necessarily have an asymptotic average velocity.",

author = "Yuri Bakhtin and Liying Li",

note = "Funding Information: Received by the editors August 16, 2018, and, in revised form, August 24, 2019. 2010 Mathematics Subject Classification. Primary 34F05, 37A25, 37A50, 60K37; Secondary 35B27. The authors were partially supported by the NSF via award DMS-1811444. Funding Information: We learned about the question that we study in this paper from Alexei Novikov. We are grateful to him, Leonid Koralov, and Arjun Krishnan for the discussions that followed. We thank the referee for useful comments. In addition, we gratefully acknowledge partial support from the NSF. Publisher Copyright: {\textcopyright} 2020 American Mathematical Society",

year = "2020",

month = nov,

doi = "10.1090/proc/15147",

language = "English (US)",

volume = "148",

pages = "4733--4744",

journal = "Proceedings of the American Mathematical Society",

issn = "0002-9939",

publisher = "American Mathematical Society",

number = "11",

}