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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2508.21658 |
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| _version_ | 1866910144611221504 |
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| author | Osada, Hirofumi Osada, Shota |
| author_facet | Osada, Hirofumi Osada, Shota |
| contents | We study the infinite-dimensional stochastic differential equations (ISDEs) of infinite-particle systems associated with Coulomb random point fields. The stochastic dynamics described by these ISDEs are referred to as Coulomb interacting Brownian motions. In all spatial dimensions $ d \ge 2 $ and for all inverse temperatures $ β> 0 $, we construct the Coulomb interacting Brownian motions.
We prove that the ISDEs admit strong solutions and that pathwise uniqueness holds. The resulting labeled dynamics form an $ \RdN $-valued diffusion, possibly without an invariant measure, while the corresponding unlabeled process is a reversible diffusion with respect to the underlying Coulomb random point field.
Moreover, we identify the infinite-particle stochastic dynamics as the limit in path space of finite-particle systems driven by stochastic differential equations. This identification is achieved through two approximation schemes: finite-domain systems with reflecting boundary conditions and $ N $-particle systems. Although the $ N $-particle approximation is more fundamental, its justification relies crucially on the finite-domain approximation together with the uniqueness of solutions to the ISDEs.
Previously, only the case $ d = 2 $ and $ β= 2 $, known as the Ginibre interacting Brownian motion, was understood through random matrix theory and determinantal random point fields. Extending this result beyond the determinantal setting has remained a major difficulty.
We introduce a new, conceptually clear method based on stochastic analysis of infinite-particle systems with long-range interactions that yields a rigorous construction of Coulomb interacting Brownian motions. A key ingredient is an explicit computation of the logarithmic derivatives of Coulomb random point fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_21658 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Infinite-dimensional stochastic differential equations for Coulomb random point fields Osada, Hirofumi Osada, Shota Probability We study the infinite-dimensional stochastic differential equations (ISDEs) of infinite-particle systems associated with Coulomb random point fields. The stochastic dynamics described by these ISDEs are referred to as Coulomb interacting Brownian motions. In all spatial dimensions $ d \ge 2 $ and for all inverse temperatures $ β> 0 $, we construct the Coulomb interacting Brownian motions. We prove that the ISDEs admit strong solutions and that pathwise uniqueness holds. The resulting labeled dynamics form an $ \RdN $-valued diffusion, possibly without an invariant measure, while the corresponding unlabeled process is a reversible diffusion with respect to the underlying Coulomb random point field. Moreover, we identify the infinite-particle stochastic dynamics as the limit in path space of finite-particle systems driven by stochastic differential equations. This identification is achieved through two approximation schemes: finite-domain systems with reflecting boundary conditions and $ N $-particle systems. Although the $ N $-particle approximation is more fundamental, its justification relies crucially on the finite-domain approximation together with the uniqueness of solutions to the ISDEs. Previously, only the case $ d = 2 $ and $ β= 2 $, known as the Ginibre interacting Brownian motion, was understood through random matrix theory and determinantal random point fields. Extending this result beyond the determinantal setting has remained a major difficulty. We introduce a new, conceptually clear method based on stochastic analysis of infinite-particle systems with long-range interactions that yields a rigorous construction of Coulomb interacting Brownian motions. A key ingredient is an explicit computation of the logarithmic derivatives of Coulomb random point fields. |
| title | Infinite-dimensional stochastic differential equations for Coulomb random point fields |
| topic | Probability |
| url | https://arxiv.org/abs/2508.21658 |