Brownian Motion Introduction

[Latarsha Dorrance]

Thisopen access textbook is the first to provide Business and Economics Ph.D. students with a precise and intuitive introduction to the formal backgrounds of modern financial theory. It explains Brownian motion, random processes, measures, and Lebesgue integrals intuitively, but without sacrificing the necessary mathematical formalism, making them accessible for readers with little or no previous knowledge of the field. It also includes mathematical definitions and the hidden stories behind the terms discussing why the theories are presented in specific ways.

Andreas Lffler received his postdoctoral qualification (habilitation) in Mathematics and Economics from the University of Leipzig and Free University Berlin, Germany, and has been a Professor of Banking and Finance at the Department of Finance, Accounting and Taxation of the Free University of Berlin since 2012.

Abstract. Stochastic processes exhibiting power-law slopes in the frequency domain are frequently well modeled by fractional Brownian motion (fBm), with the spectral slope at high frequencies being associated with the degree of small-scale roughness or fractal dimension. However, a broad class of real-world signals have a high-frequency slope, like fBm, but a plateau in the vicinity of zero frequency. This low-frequency plateau, it is shown, implies that the temporal integral of the process exhibits diffusive behavior, dispersing from its initial location at a constant rate. Such processes are not well modeled by fBm, which has a singularity at zero frequency corresponding to an unbounded rate of dispersion. A more appropriate stochastic model is a much lesser-known random process called the Matrn process, which is shown herein to be a damped version of fractional Brownian motion. This article first provides a thorough introduction to fractional Brownian motion, then examines the details of the Matrn process and its relationship to fBm. An algorithm for the simulation of the Matrn process in O(NlogN) operations is given. Unlike fBm, the Matrn process is found to provide an excellent match to modeling velocities from particle trajectories in an application to two-dimensional fluid turbulence.

In the introduction, we said that we wanted to study randomly growing surfaces, but what does that mean, exactly? What does it mean for something to be random and how can a surface grow randomly? To answer these questions, we will start more carefully and talk about random walks of particles.

Imagine a gas molecule in the air: it moves around on its own until it hits another gas molecule which makes it change direction. Since there are so many gas molecules in the air, it will constantly bump into other molecules (roughly \(10^14\) hits per second - that equals the total number of Google searches performed worldwide during 79 years!) and it will be just as likely to be hit from another particle on the left as it will be to be hit on the right. The bumps therefore cancel each other out, so after a long time interval, it will barely have moved at all, even though it makes really quick jerks all the time. The way the gas molecule moves will turn out to be important to studying randomly growing surfaces, so we will keep going on this track for a while!

a): We start with a one-dimensional motion. Draw a coordinate system with time \(t\) on the horizontal axis, and height \(h\) on the vertical axis. Mark the origin. Now, flip a coin. If heads, mark a point one step ahead and one step above the previously marked point. If tails, mark a point one step ahead and one step below the previous one. Continue this for a while and draw the resulting graph.

b): What is the average of \(h\), as a function of time? Calculate this in a table.

c): What is the average of \(h^2\), as a function of time? Calculate this in a table.

d): What way do you think would be a good way of measuring how far the random walk has gone from the origin? Is the average of \(h\) a good quantity? Is the average of \(h^2\) better or worse?

e): Next, you will draw a two-dimensional random walk. Draw a chessboard pattern around the origin, and roll a die. If you get a \(1\), mark the square to the left of the previous square. If \(2\), mark the square below the previous one. If \(3\), mark the one to the right, and if \(4\), mark the one above. If you get \(5\) or \(6\), roll again.

This is a very simple model of how the gas molecule can move, but it is also close to reality! To see a larger example, the following is a two-dimensional random walk generated in the same way as the exercise.

Real gas molecules can move in all directions, not just to neighbors on a chessboard. We would therefore like to be able to describe a motion similar to the random walk above, but where the molecule can move in all directions. A realistic description of this is Brownian motion - it is similar to the random walk (and in fact, can be made to become equal to it. See the fact box below.), but is more realistic. In the beginning of the twentieth century, many physicists and mathematicians worked on trying to define and make sense of Brownian motion - even Einstein was interested in it!

Now, Einstein realized that even though the movements of all the individual gas molecules are random, there are some quantities we can measure that are not random, they are predictable and can be calculated. One such quantity is the density \(\rho\) of the gas molecules. Einstein showed that the density satisfies a differential equation

called the diffusion equation, and where \(D\) is the diffusion coefficient that can be calculated. This is an equation that can be solved, so we are able to predict something with certainty from a random model - this is an example of the strategy that is used in statistical mechanics. Einstein's equation showed that diffusion processes, for instance seeing a drop of ink spread out in water, are caused by Brownian motion - the question we will ask for the next pages is: can Brownian motion explain also other random phenomena?

This book provides an accessible introduction to stochastic processes in physics and describes the basic mathematical tools of the trade: probability, random walks, and Wiener and Ornstein-Uhlenbeck processes. It includes end-of-chapter problems and emphasizes applications.

An Introduction to Stochastic Processes in Physics builds directly upon early-twentieth-century explanations of the "peculiar character in the motions of the particles of pollen in water" as described, in the early nineteenth century, by the biologist Robert Brown. Lemons has adopted Paul Langevin's 1908 approach of applying...

An Introduction to Stochastic Processes in Physics builds directly upon early-twentieth-century explanations of the "peculiar character in the motions of the particles of pollen in water" as described, in the early nineteenth century, by the biologist Robert Brown. Lemons has adopted Paul Langevin's 1908 approach of applying Newton's second law to a "Brownian particle on which the total force included a random component" to explain Brownian motion. This method builds on Newtonian dynamics and provides an accessible explanation to anyone approaching the subject for the first time. Students will find this book a useful aid to learning the unfamiliar mathematical aspects of stochastic processes while applying them to physical processes that he or she has already encountered.

Students will love this book. It tells them without fuss how to do simple and useful numerical calculations, with just enough background to understand what they are doing... a refreshingly brief and unconvoluted work.

[An Introduction to Stochastic Processes in Physics] presents fundamental ideas with admirable clarity and concision. The author presents in about 100 pages enough material for the student to appreciate the very different natures of stochastic and sure processes and to solve simple but important problems involving noise. Any physicist wondering what noise is about would be well advised to pack Lemons' books for their next train journey.

Self-contained and provides adequate insight into stochastic processes in physics. It is quite readable and will be useful to students interested in learning about stochastic processes and their relevance in understanding the physical phenomena. It also provides teachers a good approach to communicate the essence of the subject to students.

This is a clear, well-written, and valuable book. It is both original and important because it ties together much disparate material scattered throughout the literature into a coherent and readable form.

This book will be much appreciated by those who wish to teach, without going into excessive and demanding mathematical details, a little more than can be covered by analysing a one-dimensional random walk on a lattice or solving the Langevin equation. The author covers a lot of ground in very few pages. The last chapter, entitled 'Fluctuations without Dissipation,' gives his admirably slim volume its own flavor. I will have no hesitation in recommending the book to my students.

This is a lucid, masterfully written introduction to an often difficult subject and a text which belongs on the bookshelf of every student of statistical physics. I have every confidence that the accessibility of the presentation and the insight offered within will make it a classic reference in the field.

Professor Lemons's book has reclaimed the field of stochastic processes for physics. For too long it has been taught as a highly mathematical subject devoid of its roots in the physical sciences. Professor Lemons's book shows how the subject grew historically from early fundamental problems in physics, and how the greater minds, like Einstein, used its methods to solve problems that are still important today. The book is not only a good introduction for students, but an excellent guide for the professional.

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