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Weakly mixing smooth planar vector field without asymptotic directions


Authors: Yuri Bakhtin and Liying Li
Journal: Proc. Amer. Math. Soc. 148 (2020), 4733-4744
MSC (2010): Primary 34F05, 37A25, 37A50, 60K37; Secondary 35B27
DOI: https://doi.org/10.1090/proc/15147
Published electronically: August 11, 2020
MathSciNet review: 4143390
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Abstract | References | Similar Articles | Additional Information

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.


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Additional Information

Yuri Bakhtin
Affiliation: Courant Institute of Mathematical Sciences, New York University , 251 Mercer Street, New York, New York 10012
MR Author ID: 648835
ORCID: 0000-0003-1125-4543
Email: bakhtin@cims.nyu.edu

Liying Li
Affiliation: Courant Institute of Mathematical Sciences, New York University , 251 Mercer Street, New York, New York 10012
MR Author ID: 1293315
ORCID: 0000-0002-6640-7386
Email: liying@cims.nyu.edu

Received by editor(s): August 16, 2018
Received by editor(s) in revised form: August 24, 2019
Published electronically: August 11, 2020
Additional Notes: The authors were partially supported by the NSF via award DMS-1811444.
Communicated by: Nimish Shah
Article copyright: © Copyright 2020 American Mathematical Society