Nonlinear Filtering of Classical and Quantum Spin Systems

Abstract

In this paper we consider classical and quantum spin systems on discrete lattices and in Euclidean spaces, modeled by infinite dimensional stochastic diffusions in Hilbert spaces. Existence and uniqueness of various notions of solutions, existence and uniqueness of invariant measures as well as exponential convergence to equilibrium are known for these models. We formulate nonlinear filtering problem for these classes of models, derive nonlinear filtering equations of Fujisaki-Kallianpur-Kunita and Zakai tye, and prove existence and uniqueness of measure-valued solutions to these filtering equations. We then establish the Feller property and Markov property of the semigroups associated with the filtering equations and also prove existence and uniqueness of invariant measures. Evolution of error covariance equation for the nonlinear filter is derived. We also derive the nonlinear filtering equations associated with finitely-additive white noise formulation due to Kallianpur and Karandikar for the classical and quantum spin systems, and study existence and uniqueness of measure-valued solutions