Limit theorems for random walks with spatio-temporal drift
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917506478768128 |
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| author | Ngoc, Ngo P. N. Nguyen, Tuan-Minh |
| author_facet | Ngoc, Ngo P. N. Nguyen, Tuan-Minh |
| contents | We study a class of discrete-time random walks in $\mathbb{R}^d$ whose conditional drift decays polynomially in time and grows polynomially with the distance from the origin to the current position. This class is related to several models of self-interacting random processes. We determine the asymptotic behavior of the walk under the assumption that its increments have moments of order $p$ for some $p>2$. In the linear case, where the drift depends linearly on the current position, we establish a phase transition in the convergence in distribution of the normalized process to Gaussian limits. In the nonlinear case, we identify three distinct regimes separated by a critical line and show that the normalized process exhibits qualitatively different behaviors in each regime, including convergence in distribution to a Gaussian law, convergence to a non-Gaussian limit given by the stationary distribution of a stochastic differential equation, and almost sure localization on a hypersphere. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_17725 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Limit theorems for random walks with spatio-temporal drift Ngoc, Ngo P. N. Nguyen, Tuan-Minh Probability Statistical Mechanics Mathematical Physics 60F05, 82C41, 60G42 We study a class of discrete-time random walks in $\mathbb{R}^d$ whose conditional drift decays polynomially in time and grows polynomially with the distance from the origin to the current position. This class is related to several models of self-interacting random processes. We determine the asymptotic behavior of the walk under the assumption that its increments have moments of order $p$ for some $p>2$. In the linear case, where the drift depends linearly on the current position, we establish a phase transition in the convergence in distribution of the normalized process to Gaussian limits. In the nonlinear case, we identify three distinct regimes separated by a critical line and show that the normalized process exhibits qualitatively different behaviors in each regime, including convergence in distribution to a Gaussian law, convergence to a non-Gaussian limit given by the stationary distribution of a stochastic differential equation, and almost sure localization on a hypersphere. |
| title | Limit theorems for random walks with spatio-temporal drift |
| topic | Probability Statistical Mechanics Mathematical Physics 60F05, 82C41, 60G42 |
| url | https://arxiv.org/abs/2605.17725 |