Non-Debye relaxations: two types of memories and their Stieltjes character

Abstract: We show that spectral functions relevant for commonly used models of the non-Debye relaxation are related to the Stieltjes functions supported on the positive semiaxis. Using only this property it can be shown that the response and relaxation functions are nonnegative. They are connected to each other and obey the time evolution provided by integral equations involving the memory function $M(t)$ which is the Stieltjes function as well. This fact is also due to the Stieltjes character of the spectral function. Stochastic processes based approach to the relaxation phenomena gives possibility to identify the memory function $M(t)$ with the Laplace (L\'evy) exponent of some infinitely divisible stochastic process and to introduce its partner memory $k(t)$. Both memories are related by the Sonine equation and lead to equivalent evolution equations which may be freely interchanged in dependence of our knowledge on memories governing the process.