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Transactions of the American Mathematical Society

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Inert drift system in a viscous fluid: Steady state asymptotics and exponential ergodicity


Authors: Sayan Banerjee and Brendan Brown
Journal: Trans. Amer. Math. Soc. 373 (2020), 6369-6409
MSC (2010): Primary 60J60, 60K05; Secondary 60J55, 60H20
DOI: https://doi.org/10.1090/tran/8098
Published electronically: July 3, 2020
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Abstract: We analyze a system of stochastic differential equations describing the joint motion of a massive (inert) particle in a viscous fluid in the presence of a gravitational field and a Brownian particle impinging on it from below, which transfers momentum proportional to the local time of collisions. We study the long-time fluctuations of the velocity of the inert particle and the gap between the two particles, and we show convergence in total variation to the stationary distribution is exponentially fast. We also produce matching upper and lower bounds on the tails of the stationary distribution and show how these bounds depend on the system parameters. A renewal structure for the process is established, which is the key technical tool in proving the mentioned results.


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Additional Information

Sayan Banerjee
Affiliation: Department of Statistics and Operations Research, University of North Carolina, Chapel Hill, North Carolina 27599-3260
MR Author ID: 1029581
Email: sayan@email.unc.edu

Brendan Brown
Affiliation: Department of Statistics and Operations Research, University of North Carolina, Chapel Hill, North Carolina 27599-3260
Email: bb@live.unc.edu

DOI: https://doi.org/10.1090/tran/8098
Keywords: Reflected Brownian motion, viscosity, gravitation, local time, inert drift, renewal time, exponential ergodicity, regenerative process, total variation distance, Harris recurrence, petite sets.
Received by editor(s): May 28, 2019
Received by editor(s) in revised form: January 2, 2020
Published electronically: July 3, 2020
Additional Notes: The first author was partially supported by a Junior Faculty Development Grant made by UNC, Chapel Hill.
Article copyright: © Copyright 2020 American Mathematical Society