### Abstract

We present general theorems solving the long-standing problem of the existence and pathwise uniqueness of strong solutions of infinite-dimensional stochastic differential equations (ISDEs) called interacting Brownian motions. These ISDEs describe the dynamics of infinitely-many Brownian particles moving in R^{d} with free potential Φ and mutual interaction potential Ψ. We apply the theorems to essentially all interaction potentials of Ruelle’s class such as the Lennard-Jones 6-12 potential and Riesz potentials, and to logarithmic potentials appearing in random matrix theory. We solve ISDEs of the Ginibre interacting Brownian motion and the sine_{β} interacting Brownian motion with β= 1 , 2 , 4. We also use the theorems in separate papers for the Airy and Bessel interacting Brownian motions. One of the critical points for proving the general theorems is to establish a new formulation of solutions of ISDEs in terms of tail σ-fields of labeled path spaces consisting of trajectories of infinitely-many particles. These formulations are equivalent to the original notions of solutions of ISDEs, and more feasible to treat in infinite dimensions.

Original language | English |
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Pages (from-to) | 1137-1242 |

Number of pages | 106 |

Journal | Probability Theory and Related Fields |

Volume | 177 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - 2020 Aug 1 |

### Keywords

- Infinite-dimensional stochastic differential equations
- Interacting Brownian motions
- Pathwise uniqueness
- Random matrices
- Strong solutions

### ASJC Scopus subject areas

- Analysis
- Statistics and Probability
- Statistics, Probability and Uncertainty

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## Cite this

*Probability Theory and Related Fields*,

*177*(3-4), 1137-1242. https://doi.org/10.1007/s00440-020-00981-y