I’m currently interested in learning some topics about the Brownian motion and the random walk (in general, from a pure statistical and probabilistic way).

For that, I would like to ask you if you can recommend some books so I can learn the basic stuff and prepare myself for the real deal.

As I learn these topics, I would like to complement my knowledge about this subject by using programs to make simulations, any idea where I can learn about this?

If you have any ideas of where should I begin, please let me know and tell me the book/author. Thank you!

  • $\begingroup$ do you know any coding language? $\endgroup$ – Wolphram jonny Dec 12 '19 at 23:04
  • $\begingroup$ Hi @Wolphramjonny I’m good at working with Python and I was thinking using this language, what do you think? I accept other suggestions of course :) $\endgroup$ – user249212 Dec 12 '19 at 23:06

While you can construct mathematical models as simple as integrating white noise which evolve trajectories that have the appearance of Brownian motion, physical Brownian motion occurs due to a more complex, stochastic dynamic equilibrium between macroscopic particles and atoms/molecules. So if you are interested in real Brownian motion you need to consider the macroscopic properties such as mass, size, viscosity, geometry, etc. together with the atomic properties of the suspending fluid such as temperature, pressure, atomic weight, etc. to formulate the equation of dynamic equilibrium. That's what Einstein first figured out in his paper in Annalen der Physik which became his PhD thesis.

My recommendation is that you start there.

Beyond that more recent, interesting work models Brownian motion rather using Fractional Calculus (as opposed to the integral calculus Einstein used). You can google Bruce J West for work in this area.

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