General diffusions on metric graphs as limits of time-space Markov Chains
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908473975898112 |
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| author | Anagnostakis, Alexis |
| author_facet | Anagnostakis, Alexis |
| contents | We introduce the Space-Time Markov Chain Approximation (STMCA) for a general diffusion process on a finite metric graph $Γ$. The STMCA is a doubly asymmetric (in both time and space) random walk defined on a subdivisions of $Γ$, with transition probabilities and conditional transition times that match, in expectation, those of the target diffusion. We derive bounds on the $p$-Wasserstein distances between the diffusion and its STMCA in terms of a thinness quantifier of the subdivision. This bound shows that convergence occurs at any rate inferior to $\frac{1}{4} \wedge \frac{1}{p} $ in terms of the the maximum cell size of the subdivision, for adapted subdivisions, at any rate inferior to $\frac{1}{2} \wedge \frac{2}{p} $. Additionally, we provide explicit analytical formulas for transition probabilities and times, enabling practical implementation of the STMCA. Numerical experiments illustrate our results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_23724 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | General diffusions on metric graphs as limits of time-space Markov Chains Anagnostakis, Alexis Probability 60J60, 60J55, 35J08, 60J50 We introduce the Space-Time Markov Chain Approximation (STMCA) for a general diffusion process on a finite metric graph $Γ$. The STMCA is a doubly asymmetric (in both time and space) random walk defined on a subdivisions of $Γ$, with transition probabilities and conditional transition times that match, in expectation, those of the target diffusion. We derive bounds on the $p$-Wasserstein distances between the diffusion and its STMCA in terms of a thinness quantifier of the subdivision. This bound shows that convergence occurs at any rate inferior to $\frac{1}{4} \wedge \frac{1}{p} $ in terms of the the maximum cell size of the subdivision, for adapted subdivisions, at any rate inferior to $\frac{1}{2} \wedge \frac{2}{p} $. Additionally, we provide explicit analytical formulas for transition probabilities and times, enabling practical implementation of the STMCA. Numerical experiments illustrate our results. |
| title | General diffusions on metric graphs as limits of time-space Markov Chains |
| topic | Probability 60J60, 60J55, 35J08, 60J50 |
| url | https://arxiv.org/abs/2507.23724 |