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 {mathbb {R}}^d with free potential varPhi and mutual interaction potential varPsi . 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_{beta } 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.
Highlights
ΔX particular, our dynamics i and t for β prove that they are equal to those of [14] and others
We shall prove that infinite-dimensional stochastic differential equations (ISDEs) (1.5) and (1.6) have the same strong solutions, reflecting the dynamical rigidity of two-dimensional stochastic Coulomb systems called the Ginibre random point field
We introduce a new method of establishing the existence of strong solution and the pathwise uniqueness of solution of the ISDEs, including the
Summary
In Theorem 4.1 (First tail theorem), we shall give a sufficient condition of the existence of the strong solutions and the pathwise uniqueness in terms of the property of Ps,b-triviality of Tpath(RdN). The existence of a weak solution has been established in the first step, and we shall prove the pathwise uniqueness and the existence of strong solutions together using the analysis of the tail σ -field of the labeled path space. The difficulty in controlling Tpath(RdN) under the distribution given by the solution of ISDE (1.1) is that the labeled dynamics X = (Xi )i∈N have no associated stationary measures because they would be an approximately infinite product of Lebesgue measures (if they exist). 4, clarify the relation between a strong solution and a weak solution satisfying (IFC) and triviality of Tpath(RdN) in Theorem 4.1 We do this in a general setting beyond interacting Brownian motions.
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