Nonlinear Volterra‐type diffusion equations and theory of quantum fields

Abstract

We study nonlinear Volterra‐type evolution integral equations of the form: in a C∗‐algebra or in a Hilbert algebra of Dixmier‐Segal type, acting on a Hilbert space tensor product , where denotes a Hilbert space and is the Boson‐Einstein (Fermion‐Dirac) Fock space, over a complex Hilbert space . Under suitable Carathéodory‐type conditions on the corresponding Nemytskii operator Φ of f and assuming that k is a quantum dynamical‐type semigroup, we obtain exactly one classical global solution in the space of bounded continuous (operator‐valued) quantum stochastic processes. Moreover, we prove the existence of exactly one positive (respectively completely positive) classical global solution in (respectively in , applying a positivity (respectively completely positivity preserving) quantum stochastic integration process and assuming that k is a quantum dynamical semigroup acting on , where Φ defines a positive (respectively completely positive) quantum stochastic process.