Non-equilibrium Particle Dynamics with Unbounded Number of Interacting Neighbors

We consider an infinite system of first order differential equations in \({\mathbb {R}}^{\nu }\), parameterized by elements x of a fixed countable set \(\gamma \subset {\mathbb {R}}^d\), where the right-hand side of each x-equation depends on a finite but in general unbounded number \(n_x\) of variables (a row-finite system). Such systems describe in particular (non-equilibrium) dynamics of spins \(q_x\in {\mathbb {R}}^{\nu }\) of a collection of particles labelled by points \(x\in \gamma \). Two spins \(q_{x}\) and \(q_{y}\) interact via a pair potential if the distance between x and y is no more than a fixed interaction radius. In contrast to the case where \(\gamma \) is a regular graph, e.g. \(\mathbb {Z}^d\), the number \(n_x\) of particles interacting with particle x can be unbounded in x. Our main example of a “growing” configuration \(\gamma \) is a typical realization of a Poisson (or Gibbs) point process. Under certain dissipativity-type condition on the right-hand side of our system and a bound on growth of \(n_x\), we prove the existence and (under additional assumptions) uniqueness of infinite lifetime solutions with explicit estimates of growth in parameter x and time t. For this, we obtain uniform estimates of solutions to approximating finite systems using a version of Ovsyannikov’s method for linear systems in a scale of Banach spaces. As a by-product, we develop an infinite-time generalization of the Ovsyannikov method.