Brownian motion and diffusion: from stochastic processes to chaos and beyond.

A Statistical Study of two Diffusion Processes on Torus and Their Applications

Diffusion Processes such as Brownian motions and Ornstein-Uhlenbeck processes are the classes of stochastic processes that have been investigated by researchers in various disciplines including biological sciences. It is usually assumed that the outcomes of these processes are laid on the Euclidean spaces. However, some data in physical, chemical and biological phenomena indicate that they cann...

From diffusion to anomalous diffusion: a century after Einstein's Brownian motion.

Einstein's explanation of Brownian motion provided one of the cornerstones which underlie the modern approaches to stochastic processes. His approach is based on a random walk picture and is valid for Markovian processes lacking long-term memory. The coarse-grained behavior of such processes is described by the diffusion equation. However, many natural processes do not possess the Markovian pro...

Existence and Measurability of the Solution of the Stochastic Differential Equations Driven by Fractional Brownian Motion

Title: Anomalous Diffusion in a Weakly Chaotic Dynamical System

Background: Anomalous transport became a very important topic over the past couple of years [1]. This new, highly interdisciplinary field of research deals with deviations from simple Brownian motion in terms of non-Gaussian processes and non-Markovian dynamics. These dynamics lead to phenomena like subor superdiffusion, i.e., where particles spread out slower or faster, respectively, than comp...

On time-dependent neutral stochastic evolution equations with a fractional Brownian motion and infinite delays

In this paper, we consider a class of time-dependent neutral stochastic evolution equations with the infinite delay and a fractional Brownian motion in a Hilbert space. We establish the existence and uniqueness of mild solutions for these equations under non-Lipschitz conditions with Lipschitz conditions being considered as a special case. An example is provided to illustrate the theory