Modeling stochastic Langevin dynamics in fractal dimensions

The Langevin equation is a Newtonian equation describing the evolution of a dynamical system when subjected to a combination of deterministic and fluctuating or random forces. It is one of best-known stochastic differential equations in statistical physics and kinetic theory describing the motion of a complex dynamical system of particles perturbed by some white noise. This equation is usually used based on the assumption that the location of the particle at a moment depends only on its preceding location and not on that of long time before. Its solution is of Markov property that expresses a loss-memory evolution of the system. In this study, a fractal Langevin equation is proposed to study the random walks of particles exhibiting strange displacements driven by Gaussian white noise and memory kernel. Two different models have been introduced: local and nonlocal kernels. The first model is suitable to describe subdiffusion, whereas the second model, the dynamics exhibit random oscillations that show considerable fluctuations in frequency and amplitude. Our models show that the stochastic oscillation arises from a fractal random walk process, and prove the relevance of fractals in stochastic anomalous random walk processes. Additional features have been discussed.