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.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
S. Albeverio, Yu. G. Kondratiev, M. Röckner, and T. V. Tsikalenko. Glauber dynamics for quantum lattice systems, Rev. Math. Phys. 13 No. 1, 51–124 (2001).
G. Leha and G. Ritter. On solutions to stochastic differential equations with discontinuous drift in Hilbert space, Math. Ann. 270, 109–123 (1985).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Gawarecki, L., Mandrekar, V. (2011). Stochastic Differential Equations with Discontinuous Drift. In: Stochastic Differential Equations in Infinite Dimensions. Probability and Its Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16194-0_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-16194-0_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-16193-3
Online ISBN: 978-3-642-16194-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)