Nonlinear Volterra‐type diffusion equations and theory of quantum fields

Abstract

We study nonlinear Volterra‐type evolution integral equations of the form: xfalse(tfalse)=hfalse(tfalse)+∫0tkfalse(t,sfalse)ffalse(s,xfalse(sfalse)false)0.3emds,t0.3em∈0.3emR+ in a C∗‐algebra 𝒜 or in a Hilbert algebra of Dixmier‐Segal type, acting on a Hilbert space tensor product ℋ=scriptH⊗ℱ, where scriptH denotes a Hilbert space and ℱ is the Boson‐Einstein (Fermion‐Dirac) Fock space, over a complex Hilbert space 𝒦. 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 Cbfalse(R+,𝒜false) of bounded continuous (operator‐valued) quantum stochastic processes. Moreover, we prove the existence of exactly one positive (respectively completely positive) classical global solution in Cbfalse(R+,𝒜false) (respectively in Cbfalse(R+,ℒfalse(𝒜false)false), applying a positivity (respectively completely positivity preserving) quantum stochastic integration process and assuming that k is a quantum dynamical semigroup acting on 𝒜, where Φ defines a positive (respectively completely positive) quantum stochastic process.