MATHEMATICAL ANALYSIS OF SOME ITERATIVE METHODS FOR THE RECONSTRUCTION OF MEMORY KERNELS*

We analyze three iterative methods that have been proposed in the computational physics community for the reconstruction of memory kernels in a stochastic delay differential equation known as the generalized Langevin equation. These methods use the autocorrelation function of the solution of this equation as input data. Although they have been demonstrated to be useful, a straightforward Laplace analysis does not support their conjectured convergence. We provide more detailed arguments to explain the good performance of these methods in practice. In the second part of this paper we investigate the solution of the generalized Langevin equation with a perturbed memory kernel. We establish sufficient conditions including error bounds such that the stochastic process corresponding to the perturbed problem converges to the unperturbed process in the mean square sense.

Automated summary

This paper analyzes three iterative methods proposed in computational physics for reconstructing memory kernels in the generalized Langevin equation, providing detailed arguments for their practical performance. Additionally, it investigates the solution of the generalized Langevin equation with perturbed memory kernel and establishes sufficient conditions, including error bounds, for convergence of the stochastic process to the unperturbed process in the mean square sense.