TY - JOUR

T1 - Infinite-dimensional stochastic differential equations and tail σ -fields

AU - Osada, Hirofumi

AU - Tanemura, Hideki

N1 - Funding Information:
H.O. is supported in part by a Grant-in-Aid for Scientific Research (Grant Nos. 16K13764, 16H02149, 16H06338, and KIBAN-A, No. 24244010) from the Japan Society for the Promotion of Science. H.T. is supported in part by a Grant-in-Aid for Scientific Research (KIBAN-C, No. 15K04916, Scientific Research (B), No. 19H01793) from the Japan Society for the Promotion of Science.
Publisher Copyright:
© 2020, The Author(s).

PY - 2020/8/1

Y1 - 2020/8/1

N2 - 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 Rd 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 sineβ 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.

AB - 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 Rd 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 sineβ 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.

KW - Infinite-dimensional stochastic differential equations

KW - Interacting Brownian motions

KW - Pathwise uniqueness

KW - Random matrices

KW - Strong solutions

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U2 - 10.1007/s00440-020-00981-y

DO - 10.1007/s00440-020-00981-y

M3 - Article

AN - SCOPUS:85087561427

SN - 0178-8051

VL - 177

SP - 1137

EP - 1242

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

IS - 3-4

ER -