Time deformations of master equations

We subject convolutionless and convolution master equations to time deformations and explore properties of the modified maps. We obtain the following results: (i) the original convolutionless master equation is shown to describe a completely positive divisible dynamics if and only if the deformed map is completely positive under any time deformation; if the deformed map is not completely positive, then the original dynamics is at least weakly non-Markovian; (ii) we find a necessary condition for positive divisibility of a Hermitian commutative dynamical map given by a convolution master equation; (iii) proper time deformations of the memory kernel for Pauli dynamical qubit maps preserve positivity of the deformed map if the original convolution master equation describes a positive divisible process; if the deformed map is not positive, then the original qubit dynamics is essentially non-Markovian. We provide examples of time-local and convolution master equations for qubits to illustrate the results.