Heterogeneous position jump process and diffusion laws as result

Abstract

that is, a heterogeneous random walk can have various dynamics depending on the particle taking walk-length at either departure point or arrivale point. This dissertation develops several models from a discrete velocity kinetic model or general version of random walk or position jump process and nonlocal diffusion equation and derives a correct form of diffusion equation with convergence of the solutions.

The first part of this paper models a heterogeneous diffusion equation from a discrete velocity kinetic model. The discrete kinetic model is defined on a $n$-dimensional cube with a periodic boundary, with $2n$ fractional species, with each species having a speed as a function of position and direction normal to each face of the cube. The kinetic model is well-defined, known from the classical semigroup theory and is applied a parabolic scaling to give one parameter family of the kinetic system. An Energy functional is defined and used to give weighted $L^p$ estimate of the kinetic system by its initial condition and the uniform $L^p$ estimate gives a weak subsequential convergence. Finally, Div-Curl lemma is used to conclude a strong convergence of solutions of the parametrized kinetic system to a unique limit, which is the solution of the resulting heterogeneous diffusion equation.

The second part models a heterogeneous diffusion by a position jump process. It is known in the community that random walk can be defined and numerically simulated with a nonconstant walk-length and sojourn time. Different from homogeneous random walk, there is another parameter called a reference point, which means at each jump a particle may choose the value of walk-length or sojourn time of the arrival point or the departing point. It is known that the numerical result quite varies when the reference is changed for the walk-length but stays the same for the sojourn time. This research proposes a discrete-time continuous space model that answers those previous results and derives a family of heterogeneous diffusion equation as a result. First, a generic recurrence relation is defined that models a simple random walk with a heterogeneous walk-length along a discrete time set. Then jump operators are defined which describe position jumps with heterogeneity. By a parabolic scale limit, one can calculate resulting diffusion equation that varies with a reference point parameter. With comparison principle established, we prove a uniform convergence of solutions for recursive relation to the solution of the local diffusion equation. Finally, a simple argument is proposed to understand that the sojourn time does not work in the way as the walk-length and the same uniform convergence and final diffusion equation are shown from the updated model that includes a heterogeneity in both sojourn time and walk-length.

The last part discusses another position jump process that is continuous time model while the previous one models a discrete time process. The model begins from the known equation from the continuous time Markov chain of the probability theory. With this knowledge, we model a nonlocal diffusion equation that incorporates both heterogeneous departing rate and walk-length where the departing rate is described as the inverse of the sojourn time. Then the existence and uniqueness of the nonlocal diffusion equation is shown by the Banach fixed point theorem and also comparison is checked. One can calculate to derive local diffusion equation as a parabolic scale limit and can show the uniform convergence with the comparison principle.

This thesis discusses and models heterogeneous diffusion phenomenon. In general within a heterogeneous environment, a species of particles do not diffuse in a way to uniformize its density. This heteregeneous diffusion problem arose from mathematical biology to explain the dynamics, steady distribution and diffusion of organisms within given environment. This kind of heterogeneity is of a great interest in various fields including mathematical bioilogy and modeling population dynamics and requires a new and generalized form of diffusion equation. While the most famous diffusion, the heat equation, can be derived from the simple random walk, one has to develop a heterogeneous version of simple model as a velocity jump process and a position jump process, both suggested by several decades ago. Another problem of reference point issue is proposed quite recently and some paper has been published to explain the issue