General diffusions on the star graph as time-changed Walsh Brownian motion
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915419213791232 |
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| author | Anagnostakis, Alexis |
| author_facet | Anagnostakis, Alexis |
| contents | We establish the representation of general regular diffusions on star-shaped graphs as time-changed Walsh Brownian motions. These are regular continuous Markov processes described locally by a family generalized second order differential operators defined on every edge and a gluing condition at the junction vertex. This allows us to prove two additional results: (i) A representation of diffusions with sticky gluing conditions as time-changes of diffusions governed by the same differential operators but with non-sticky gluing conditions. (ii) An occupation times formula for such diffusions, analogous to the classical Itô--McKean formula for one-dimensional diffusions.
Additionally, we prove two results of independent interest. First, conditions under which a diffusion on the star graph is Feller and Feller--Dynkin, extending classical results for one-dimensional diffusions. Second, the existence uniqueness of solutions to the Dirichlet problem on the unit disk of the star graph for a general diffusion operator and explicit expressions for its solution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_19299 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | General diffusions on the star graph as time-changed Walsh Brownian motion Anagnostakis, Alexis Probability 60J60, 60J55, 35J08, 60J50 We establish the representation of general regular diffusions on star-shaped graphs as time-changed Walsh Brownian motions. These are regular continuous Markov processes described locally by a family generalized second order differential operators defined on every edge and a gluing condition at the junction vertex. This allows us to prove two additional results: (i) A representation of diffusions with sticky gluing conditions as time-changes of diffusions governed by the same differential operators but with non-sticky gluing conditions. (ii) An occupation times formula for such diffusions, analogous to the classical Itô--McKean formula for one-dimensional diffusions. Additionally, we prove two results of independent interest. First, conditions under which a diffusion on the star graph is Feller and Feller--Dynkin, extending classical results for one-dimensional diffusions. Second, the existence uniqueness of solutions to the Dirichlet problem on the unit disk of the star graph for a general diffusion operator and explicit expressions for its solution. |
| title | General diffusions on the star graph as time-changed Walsh Brownian motion |
| topic | Probability 60J60, 60J55, 35J08, 60J50 |
| url | https://arxiv.org/abs/2502.19299 |