Generalized Fokker–Planck equation with generalized interval probability

The Fokker–Planck equation is widely used to describe the time evolution of stochastic systems in drift-diffusion processes. Yet, it does not differentiate two types of uncertainties: aleatory uncertainty that is inherent randomness and epistemic uncertainty due to lack of perfect knowledge. In this paper, a generalized differential Chapman–Kolmogorov equation based on a new generalized interval probability theory is derived, where epistemic uncertainty is modeled by the generalized interval while the aleatory one is by the probability measure. A generalized Fokker–Planck equation is proposed to describe drift-diffusion processes under both uncertainties. A path integral approach is developed to numerically solve the generalized Fokker–Planck equation. The resulted interval-valued probability density functions rigorously bound the real-valued ones computed from the classical path integral method. The method is demonstrated by numerical examples.

► Stochastic systems are described by a new generalized interval probability theory.
► A generalized differential Chapman–Kolmogorov equation is derived.
► A generalized Fokker–Planck equation is proposed for drift-diffusion processes.
► A path integral method is developed to model time evolution of interval probability.
► Interval-valued probability densities rigorously bound the real-valued solutions.