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Recurrence and transience of random walks with drift ρxα/tβ

  • Ngo P. N. Ngoc
  • Tuan-Minh Nguyen
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Abstract

Menshikov and Volkov [Electron. J. Probab. 13 (2008)] studied recurrence and transience of a class of Markovian random walks on ℝ+ whose conditional drift depends on both time and position and is of order ρxαt−β with ρ>0. The case on the critical line 2β−α=1, with α∈(−1,1)∖{0}, remained open. We prove recurrence in this remaining case. Furthermore, we establish recurrence and transience criteria that complete the classification for −1<α<1 and β≥ 0, without assuming the Markov property and under weaker assumptions on the increments than those imposed by Menshikov and Volkov.

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Cite

BibTeXRIS
BibTeX
@misc{ngoc2026recurrence,
  title = {{Recurrence and transience of random walks with drift $ρx^α/t^β$}},
  author = {Ngo P. N. Ngoc and Tuan-Minh Nguyen},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.11046},
  eprint = {2609.11046},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.11046},
}
RIS
TY  - UNPB
TI  - Recurrence and transience of random walks with drift ρx^α/t^β
AU  - Ngo P. N. Ngoc
AU  - Tuan-Minh Nguyen
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.11046
UR  - https://arxiv.org/abs/2609.11046
AB  - Menshikov and Volkov [Electron. J. Probab. 13 (2008)] studied recurrence and transience of a class of Markovian random walks on ℝ_+ whose conditional drift depends on both time and position and is of order ρx^αt^−β with ρ>0. The case on the critical line 2β−α=1, with α∈(−1,1)∖{0}, remained open. We prove recurrence in this remaining case. Furthermore, we establish recurrence and transience criteria that complete the classification for −1<α<1 and β≥ 0, without assuming the Markov property and under weaker assumptions on the increments than those imposed by Menshikov and Volkov.
ER  -