Analysis of an Inhomogeneous Random Walk with Spatial Decay of Transition Probabilities and Parameter Renewal per Excursion
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911621590286336 |
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| author | Yoshida, Naohiro |
| author_facet | Yoshida, Naohiro |
| contents | In this paper, we propose and analyze a novel one-dimensional inhomogeneous random walk model that combines spatial decay of transition probabilities with a temporal renewal structure for each excursion. In this model, the probability of moving to the right from each state creats a spatial inhomogeneity that causes a stronger pull-back toward the origin as the process moves farther away. Furthermore, it features a random environment aspect where the parameter of each transition probability is independently resampled from a uniform distribution at the beginning of each excursion. We rigorously derive the hitting probability to an upper boundary using a scale function. Furthermore, by solving linear difference equations, we provide the probability generating function of the first hitting time, the expected occupation time for each state during an excursion (discrete Green's function), and the distribution and expectation of the maximum penetration depth. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_22644 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Analysis of an Inhomogeneous Random Walk with Spatial Decay of Transition Probabilities and Parameter Renewal per Excursion Yoshida, Naohiro Probability 60G5, 60J10, 60K37 In this paper, we propose and analyze a novel one-dimensional inhomogeneous random walk model that combines spatial decay of transition probabilities with a temporal renewal structure for each excursion. In this model, the probability of moving to the right from each state creats a spatial inhomogeneity that causes a stronger pull-back toward the origin as the process moves farther away. Furthermore, it features a random environment aspect where the parameter of each transition probability is independently resampled from a uniform distribution at the beginning of each excursion. We rigorously derive the hitting probability to an upper boundary using a scale function. Furthermore, by solving linear difference equations, we provide the probability generating function of the first hitting time, the expected occupation time for each state during an excursion (discrete Green's function), and the distribution and expectation of the maximum penetration depth. |
| title | Analysis of an Inhomogeneous Random Walk with Spatial Decay of Transition Probabilities and Parameter Renewal per Excursion |
| topic | Probability 60G5, 60J10, 60K37 |
| url | https://arxiv.org/abs/2604.22644 |