Stochastic Differential Equations with Discontinuous Drift

Abstract

The study of interacting particle systems arise in physics in case of spin systems and behavior of systems following Glauber dynamics. They can be modeled with stochastic differential equations in ℝ^∞ or in ${\mathbb{R}}^{{\mathbb{Z}}^{d}}$, d>1. One wishes to know if a solution exists and determine its state space. For example, it may be interesting if the solution is in l ₂. However, we consider models where the drift coefficient is not continuous on l ₂. This complication, together with the fact that the Peano theorem fails in infinite dimensions, can be overcome by noting that the embedding from l ₂ to ℝ^∞ is continuous and compact, and by applying the “method of compact embedding” from Chap. 3. We construct weak solutions, generally in a Hilbert space H, which are not continuous as functions into the state space with its natural topology. However, by identifying H with l ₂↪ℝ^∞, they turn out to be continuous in the topology induced on H from ℝ^∞. This part is related to the work of Leha and Ritter (Math. Ann. 270:109–123, 1985), where equations in general form are studied and the motivation comes from modeling of unbounded spin systems. We show the existence of solutions under weaker condition on the drift and provide Galerkin approximation in the case studied by Leha and Ritter. In the second part we prove existence of solutions for quantum lattice systems in ${\mathbb{R}}^{{\mathbb{Z}}^{d}}$ under weaker assumptions than those considered by Albeverio et al. (Rev. Math. Phys. 13(1):51–124, 2001) in justifying Glauber dynamics. To that end, we change the set-up to equations in a dual to a nuclear space and obtain weak solutions using compact embeddings.