Nonlinear Volterra‐type diffusion equations and theory of quantum fields

We study nonlinear Volterra‐type evolution integral equations of the form: x ( t ) = h ( t ) + ∫ 0 t k ( t , s ) f ( s , x ( s ) ) d s , t ∈R + in a C ∗ ‐algebra or in a Hilbert algebra of Dixmier‐Segal type, acting on a Hilbert space tensor product ℋ = H ⊗ ℱ , where H denotes a Hilbert space and ℱ 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 C b ( R + , ) of bounded continuous (operator‐valued) quantum stochastic processes. Moreover, we prove the existence of exactly one positive (respectively completely positive) classical global solution in C b ( R + , ) (respectively in C b ( R + , ℒ ( ) ) , 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.