Subdiffusion through Switching
An ongoing effort in the study of microparticle movement in biofluids is the proper characterization of subdiffusive processes i.e. processes whose mean-squared displacement scales as a sublinear power law. In order to describe phenomena that lead to subdiffusive behavior, a few models have been developed: fractional Brownian motion, the generalized Langevin equation, and random walks with dependent increments. We will present perhaps a simpler model that leads to subdiffusion and is designed to characterize systems where a regularly diffusive particle intermittently becomes trapped for long periods of time. By combining ideas from Hybrid Switching Diffusion and queuing systems literature we will describe the law of our process. The major obstacle is the introduction of heavy tail immobilization times and we will overcome it by representing the power law as an infinite mixture of exponentials. The description of the law allows us also to solve the First Passage Problem. Modeling subdiffusion is a very active field of research both in mathematics and physics. Physicists often use a continuous model that originates in the theory of random walks - Brownian motion inversely subordinated to an $\alpha$-stable process. In a similar way we will describe our process. With this description we will show that our process under rescaling is equivalent to the inverse subordinated Brownian motion, i.e., we will present the functional limit theorem for Switching Diffusion.