Reduced Dynamics with Initial Correlations, and Time-Dependent Environment and Hamiltonians

We consider a small system s in a bath b, in the case that the state of b and all Hamiltonians are possibly time dependent. We obtain for the reduced density matrix of s an exact evolution equation ρₛ(t)0ex0ex=0ex0exΛ(t,τ)ρₛ(τ), with $({∂}/{∂}t){Λ}(t,{τ}){{0ex}{0ex}}={{0ex}{0ex}}{γ}(t,{τ}){Λ}(t,{τ})$, where ${γ}(t,{τ})$ depends on the system-bath correlations at time ${τ}$. The open evolutor ${Λ}(t,{τ})$ can (but need not) be chosen completely positive. It is argued that as $t{-}{τ}$ increases, ${γ}(t,{τ}){→}{{γ}}₀(t)$ forgets the initial correlations and tends to Lindblad form in time-coarse-grained weak coupling limits.