## Weakly mixing smooth planar vector field without asymptotic directions

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- by Yuri Bakhtin and Liying Li PDF
- Proc. Amer. Math. Soc.
**148**(2020), 4733-4744 Request permission

## 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.## References

- Antonio Auffinger, Michael Damron, and Jack Hanson,
*50 years of first-passage percolation*, University Lecture Series, vol. 68, American Mathematical Society, Providence, RI, 2017. MR**3729447**, DOI 10.1090/ulect/068 - Yuri Bakhtin,
*Inviscid Burgers equation with random kick forcing in noncompact setting*, Electron. J. Probab.**21**(2016), Paper No. 37, 50. MR**3508684**, DOI 10.1214/16-EJP4413 - Yuri Bakhtin, Eric Cator, and Konstantin Khanin,
*Space-time stationary solutions for the Burgers equation*, J. Amer. Math. Soc.**27**(2014), no. 1, 193–238. MR**3110798**, DOI 10.1090/S0894-0347-2013-00773-0 - V. Bergelson and A. Gorodnik,
*Weakly mixing group actions: a brief survey and an example*, Modern dynamical systems and applications, Cambridge Univ. Press, Cambridge, 2004, pp. 3–25. MR**2090763** - Yuri Bakhtin and Konstantin Khanin,
*On global solutions of the random Hamilton-Jacobi equations and the KPZ problem*, Nonlinearity**31**(2018), no. 4, R93–R121. MR**3816628**, DOI 10.1088/1361-6544/aa99a6 - Jon Chaika and Arjun Krishnan,
*Stationary coalescing walks on the lattice*, Probab. Theory Related Fields**175**(2019), no. 3-4, 655–675. MR**4026602**, DOI 10.1007/s00440-018-0893-2 - Eric Cator and Leandro P. R. Pimentel,
*A shape theorem and semi-infinite geodesics for the Hammersley model with random weights*, ALEA Lat. Am. J. Probab. Math. Stat.**8**(2011), 163–175. MR**2783936** - Pierre Cardaliaguet and Panagiotis E. Souganidis,
*Homogenization and enhancement of the $G$-equation in random environments*, Comm. Pure Appl. Math.**66**(2013), no. 10, 1582–1628. MR**3084699**, DOI 10.1002/cpa.21449 - Olle Häggström and Ronald Meester,
*Asymptotic shapes for stationary first passage percolation*, Ann. Probab.**23**(1995), no. 4, 1511–1522. MR**1379157** - C. Douglas Howard and Charles M. Newman,
*Geodesics and spanning trees for Euclidean first-passage percolation*, Ann. Probab.**29**(2001), no. 2, 577–623. MR**1849171**, DOI 10.1214/aop/1008956685 - Wenjia Jing, Panagiotis E. Souganidis, and Hung V. Tran,
*Large time average of reachable sets and applications to homogenization of interfaces moving with oscillatory spatio-temporal velocity*, Discrete Contin. Dyn. Syst. Ser. S**11**(2018), no. 5, 915–939. MR**3817561**, DOI 10.3934/dcdss.2018055 - Cristina Licea and Charles M. Newman,
*Geodesics in two-dimensional first-passage percolation*, Ann. Probab.**24**(1996), no. 1, 399–410. MR**1387641**, DOI 10.1214/aop/1042644722 - James Nolen and Alexei Novikov,
*Homogenization of the G-equation with incompressible random drift in two dimensions*, Commun. Math. Sci.**9**(2011), no. 2, 561–582. MR**2815685** - Fraydoun Rezakhanlou and James E. Tarver,
*Homogenization for stochastic Hamilton-Jacobi equations*, Arch. Ration. Mech. Anal.**151**(2000), no. 4, 277–309. MR**1756906**, DOI 10.1007/s002050050198 - Panagiotis E. Souganidis,
*Stochastic homogenization of Hamilton-Jacobi equations and some applications*, Asymptot. Anal.**20**(1999), no. 1, 1–11. MR**1697831** - Mario V. Wüthrich,
*Asymptotic behaviour of semi-infinite geodesics for maximal increasing subsequences in the plane*, In and out of equilibrium (Mambucaba, 2000) Progr. Probab., vol. 51, Birkhäuser Boston, Boston, MA, 2002, pp. 205–226. MR**1901954**

## 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
- © Copyright 2020 American Mathematical Society
- 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
- MathSciNet review: 4143390