Deriving Stochastic Processes from Fractional Calculus Equations

This is a topic I am very excited about. This is a one hundred percent hobby project, something that I think about in my spare time because it is fun and could potentially help a lot of people. I want others to be excited about this too and learn, and maybe even contribute and collaborate. So, come over here and sit with me for a moment, let's go over some awesome math together. :)

This is meant to be a basic introduction, with some resources attached to allow you to go deeper. If there's enough interest, I may do follow up posts exploring concepts in depth.

What's a stochastic process?

It is an equation that can give you the ability to predict what will happen in the future even when you aren't quite certain about all the values in your system. I first stumbled upon it in college when I saw it being used to predict future stock prices and stock market states. But it is also essential in fields like quantum mechanics, where you definitely aren't sure exactly where your super-tiny crazy particles are and yet need to predict their motions and behaviors. And think about it-- what do we really know, for sure?

Can we use this to model feelings? Social systems? Models of political structures? Business management models?

What other subjects involve systems and simulations in which you aren't quite sure what the values are, but have a guess? What if this could be used as an optimization technique, when figuring out the precise values is just too costly?

Can we use stochastic processes to model... all physical phenomena? Everything?

And that's where this starts to get really exciting. Stochastic analysis is a rapidly growing field changing the way we are viewing math in general. We are in the midst of a potential mathematical revolution, and many people don't realize it.

How on Earth can we model physical phenomena using stochastic analysis?

Good, glad you're asking. Let's take a step back and think about something else for a moment.

I'm a graphics programmer. I love low-level graphics, and a big part of that is thinking about innovation in terms of simulation. Want beautiful cloth on your characters? Water? Hair? Particle systems that create realistic smoke and fire? All those systems can depend on a physics simulation, and it's sometimes quite complex. Graphics programmers become especially interested when that simulation also runs on the GPU.

Look at this: http://graphics.ucmerced.edu/~oozgen/paper-tog-10/

That is a paper on underwater cloth simulation that uses the concept of fractional calculus.

Fractional calculus allows us to perform integration and derivation on non-integer orders. What does that mean, on a basic level? Velocity is the first derivative of position with respect to time-- in other words, it's how position changes over time. Acceleration is the second derivative of position with respect to time-- in other words, it's how velocity changes over time. What is the half derivative of position with respect to time? What about an imaginary order? What about a negative order? What can we do with mathematical manipulation when we aren't limited to integer-based derivation and integration, and what does that imply?

Adding a fractional term in this particular underwater cloth simulation allowed for the researchers to simulate the history-laden interactions involving complex physical simulation. They were able to produce great results with fractional calculus simulation.

Can we derive stochastic processes from those fractional calculus equations, allowing for potential deterministic simulation, more efficient simulation, and more opportunity for different mathematical manipulation and analysis? Deterministic physics simulation alone would be a staggering discovery. Right now much of complex physics simulation relies on running your simulation each timestep that happened before your current one. Want to know where your water will be in 5 years? Enjoy the wait. We could potentially tell you near-instantly with stochastic analysis.

I'm going to say: yes. I attended a talk at the London Mathematical Society in which a tiny research group had derived stochastic processes successfully from appropriately constrained fractional calculus equations (https://www.maths.ox.ac.uk/events/conferences/women-maths , I'll have to dig up the name). And upon further research, I see this is now becoming more and more common (https://www.google.com/?gws_rd=ssl#q=deriving+stochastic+processes+from+fractional+derivatives). I even see people already deriving stochastic processes from fractional calculus equations (http://citeseerx.ist.psu.edu/viewdoc/downloaddoi=10.1.1.1.1891&rep=rep1&type=pdf).  And I am seeing research on deterministic equations for certain stochastic processes (http://arxiv.org/abs/cond-mat/0210373).

This is all brand new research.

It is exciting.

There's lots to discover.

I wanted to give you a taste of it.

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