Author: P.Shiu. (bizquant@gmail.com)

 

Prelude Brownian Motion is often viewed as the description of observed particles randomly colliding with surrounding particles. And Stochastic process is typically defined as follows: $$dx=\mu dt+\sigma dW$$ It is suggesting that Brownian particles randomly collide around a deterministic mean that can change over time. It is however, in violation of Newtonian rules, as particles would not spontaneously know where the mean is, unless some form of force field/velocity gradient are present in the system. This article proceeds to model random walks with the potential energy adjustments that can be used to describe a finite entropy system. The diffusion version (1st order) governing PDE is in a form of Reaction-Diffusion process. In the paper, we will lay out the foundation by modeling Brownian particles randomly colliding (diffusion) but pulled to each other by force to deterministic mean in a vacuum, simulated as graph below: The Stochastic process describing such system should be a Jump-to-mean process where jump arrival rate is determined by spatial velocity gradient, and d$\xi$ term is Laplace distribution with variance one. $$dx=\left(\mu -X\right)dJ+\sigma d\xi$$ The results can also been seen in the financial system, as shown in the daily stock price change density function on the right. It is much closer to Laplace distribution than normal distribution. More details of stock market properties are discussed in Chapter 5 - Financial Market Application . A 2nd order version PDE with wave-diffusion properties is in Chapter 6 - Wave-Particle Duality. Such system is both Newtonian & Relativity compatible, at the expense of assuming away the need for time. Time, in this construct, is just measurement indexed off Energy or Space. Please do not hesitate to ask Author for any questions. Constructive feedbacks are always greatly appreciated. Shared files Paper.pdf (Not peer-reviewed) Finite Difference.xlsm Finite Difference Wave Particle.xlsm Density Function.xlsm Below are a few chapters to outline some discussions mostly in the paper, but with some additional examples to establish fundamental concepts. PDE Summary (Click to enlarge graph) Density Function Summary (Click to enlarge graph) Contents Chapter 1 - Variance and Kinetic Energy Chapter 2 - Arrival rate and Velocity Gradient Chapter 3 - Variance + Arrival = Kinetic + Potential Energy Chapter 4 - Equilibrium State Chapter 5 - Financial Market Application Chapter 6 - Wave-Particle Duality

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Disclaimer: All contents on this website have not been peer-reviewed, and it may contain errors. It is strictly the author’s personal ideas that do not reflect any organization’s view. Furthermore, it is in disagreement with generally accepted practice of Brownian process. The proposal is original to the best of author’s knowledge. The author also reserves all the rights to the contents, including but not limited to, any copy rights, rights to modify and remove any statements, and the right to distribution. Any application of the model is used at the readers own risks.