An infinite system of Brownian balls with infinite range interaction.
The authors consider a system of infinitely many indistinguishable hard balls in a \(d\)-dimensional Euclidean space \(\mathbb R^d, d\geq 2\), undergoing Brownian motions and submitted to the influence of a smooth infinite range pair potential \(\Phi _s\). The spatial mean density of the initial configuration is allowed to be arbitrarily large and the balls may interact even if the distance between them is large. The system is represented as a unique strong solution to the infinite-dimensional Skorokhod-type system of equations \[ X_i(t)=X_i+ B_i(t)-\frac {1}{2}\sum _{j\in N}\int _0^t\nabla \Phi _s(X_i(s)-X_j(s))\,ds +\sum _{j\in N}\int _0^t(X_i(s)-X_j(s))\,dL_{ij}(s), \] where \(\{L_{ij}\}\) are the respective local times. Also, some geometrical aspect of the configuration space is discussed and the dynamics of finitely many hard balls is solved. The reversible solution of the equation is shown to be the limit of finite-dimensional approximations.
scientific article; zbMATH DE number 2102340