Feynman–Kac equation for Brownian non-Gaussian polymer diffusion

Abstract

The motion of the polymer center of mass (CM) is driven by two stochastic terms that are Gaussian white noise generated by standard thermal stirring and chain polymerization processes, respectively. It can be described by the Langevin equation and is Brownian non-Gaussian by calculating the kurtosis. We derive the forward Fokker–Planck equation governing the joint distribution of the motion of CM and the chain polymerization process. The backward Fokker–Planck equation governing only the probability density function (PDF) of CM position for a given number of monomers is also derived. We derive the forward and backward Feynman–Kac equations for the functional distribution of the motion of the CM, respectively, and present some of their applications, which are validated by a deep learning method based on backward stochastic differential equations (BSDEs), i.e. the deep BSDE method.