WIAS Preprint No. 2496, (2018)

Memory equations as reduced Markov processes


  • Stephan, Holger
    ORCID: 0000-0002-6024-5355
  • Stephan, Artur
    ORCID: 0000-0001-9871-3946

2010 Mathematics Subject Classification

  • 34K06 60J27 00A71 34D05 44A10 26C15


  • Markov generator, delay equation, Markov process without detailed balance, modeling memory equations, exponential kernel, reservoirs, quasiparticles, linear differential equations, Lagrange polynomial, Laplace transform, asymptotic behavior, simplex integrals, integro-differential equation, ordinary differential equations, meromorphic functions, non-autonomous, functional differential equation




A large class of linear memory differential equations in one dimension, where the evolution depends on the whole history, can be equivalently described as a projection of a Markov process living in a higher dimensional space. Starting with such a memory equation, we give an explicit construction of the corresponding Markov process. From a physical point of view the Markov process can be understood as the change of the type of some quasiparticles along one-way loops. Typically, the arising Markov process does not have the detailed balance property. The method leads to a more realisitc modeling of memory equations. Moreover, it carries over the large number of investigation tools for Markov processes to memory equations, like the calculation of the equilibrium state, the asymptotic behavior and so on. The method can be used for an approximative solution of some degenerate memory equations like delay differential equations.

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