An inverse problem for fractional random walks on finite graphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918437068996608 |
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| author | Covi, Giovanni Lassas, Matti |
| author_facet | Covi, Giovanni Lassas, Matti |
| contents | We study an inverse problem on a finite connected graph G = (X, E), on whose vertices a conductivity γ is defined. Our data consists in a sequence of partial observations of a fractional random walk on G. The observations are partial in the sense that they are limited to a fixed, observable subset B of X, while the random walk is fractional in the sense that it allows long jumps with a probability P decreasing as a fractional power of the distance along the graph. The transition probability P also depends on γ. We show that this kind of random walk data allows for the determination of a gauge class to which the transition probability matrix P belongs, which we discuss. Moreover, we show that if the transition probability matrix P is itself known, then the amount of vertices |X|, the edge set E and the conductivity γ (up to a positive factor) can be recovered. We also show a characterization of the random walk data in terms of the corresponding transition matrices P , which highlights a new surprising nonlocal property. This work is motivated by the recent strong interest in the study of the fractional Calderón problem in the Riemannian setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_05072 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An inverse problem for fractional random walks on finite graphs Covi, Giovanni Lassas, Matti Analysis of PDEs Probability 35R30, 26A33, 42B37, 46F12 We study an inverse problem on a finite connected graph G = (X, E), on whose vertices a conductivity γ is defined. Our data consists in a sequence of partial observations of a fractional random walk on G. The observations are partial in the sense that they are limited to a fixed, observable subset B of X, while the random walk is fractional in the sense that it allows long jumps with a probability P decreasing as a fractional power of the distance along the graph. The transition probability P also depends on γ. We show that this kind of random walk data allows for the determination of a gauge class to which the transition probability matrix P belongs, which we discuss. Moreover, we show that if the transition probability matrix P is itself known, then the amount of vertices |X|, the edge set E and the conductivity γ (up to a positive factor) can be recovered. We also show a characterization of the random walk data in terms of the corresponding transition matrices P , which highlights a new surprising nonlocal property. This work is motivated by the recent strong interest in the study of the fractional Calderón problem in the Riemannian setting. |
| title | An inverse problem for fractional random walks on finite graphs |
| topic | Analysis of PDEs Probability 35R30, 26A33, 42B37, 46F12 |
| url | https://arxiv.org/abs/2408.05072 |