Fokker–Planck equations on homogeneous time scales: well-posedness, stability, and applications to stochastic processes

Authors

DOI:

https://doi.org/10.31489/2026m3/170-184

Keywords:

Fokker–Planck equations, homogeneous time scales, reflective boundary conditions, abstract Cauchy problem, backward Fokker–Planck operator, asymptotic stability, Ornstein–Uhlenbeck process, geometric Brownian motion

Abstract

We study one-dimensional Fokker–Planck equations on homogeneous time scales with spatially dependent drift and diffusion coefficients. The model is formulated under reflective boundary conditions and is transformed, through the stationary density, into an abstract Cauchy problem governed by the backward Fokker–Planck operator in a weighted Hilbert space. Using semigroup methods on time scales, we prove the existence and uniqueness of mild solutions and characterize when such solutions are classical. The relation between the forward Fokker–Planck operator and the backward operator is clarified through a conjugacy with the stationary profile, which also preserves the spectral structure. Under the stated assumptions, the backward operator is positive, self-adjoint, and has a discrete spectrum. These properties yield well-posedness of the time-scale Fokker–Planck problem and asymptotic stability of solutions toward the normalized stationary density. The homogeneous time-scale setting covers both continuous and discrete dynamics in a unified framework and allows spectral representations of the resulting evolution. The applicability of the theory is illustrated by the Ornstein–Uhlenbeck process and the geometric Brownian motion, showing how classical stochastic models can be represented within the proposed time-scale formulation.

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Published

30.09.2026

How to Cite

Sutrima, S., Wibowo, S., & Sutanto, S. (2026). Fokker–Planck equations on homogeneous time scales: well-posedness, stability, and applications to stochastic processes. Bulletin of the Karaganda University. Mathematics Series, 3(123), 170–184. https://doi.org/10.31489/2026m3/170-184

Issue

No. 3 (123)/2026

Section

MATHEMATICS