Infinite-dimensional stochastic differential equations and tail σ-fields

• 成果类型: Article
• 署名作者: Osada, Hirofumi; Tanemura, Hideki
• 署名单位: Kyushu University; Keio University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-020-00981-y
• 发表日期: 2020
• 页码: 1137-1242
• 关键词: wiener-processes

摘要

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 Phi and mutual interaction potential Psi. 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 sinefi interacting Brownian motion with beta = 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 sigma-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.