Deriving continuous Brownian Motion from discrete Brownian motion.

[cosine]

Hey guys, long time no post! In the past few years I wound up getting a BA in math and then accidently wound up going to grad school. I'm giving an informal presentation in a few days where I will introduce Brownian motion to some colleagues (no laxitive jokes, please!).

 

Anyway, for the presentation I want to pass from discrete Brownian motion to continuous Brownian motion, but I need some help connecting the dots! I'll put what I have so far here and then maybe you can point the way. Thanks!

 

Here we go:

 

Here is a diagram of the simplest Brownian motion:

 

      1
  / 
0
  \
    -1

Where each possible outcome (either 1 or -1) has a probability of 1/2. Let's say this happens over the discrete time period /\s. Now let's look at two time steps:

 

          2
       /
     1
  /    \
0         0
  \    /
    -1
    |  \
    |   -2
    |    |
   [u]/\[/u]s, 2[u]/\[/u]s

 

Now we can define p(t, x), a probability density function, where t is the time, and x is the position on the vertical axis. There is a counting arguement where you define n, the number of time steps that have passed, as t//\s, that allows you to define this function explicitly. I won't go into the arguement here, but instead just show that:

 

p(t, x) = {n}CHOOSE{(n+x)/2} / 2^n

 

Ok, so this part has been straight forward. What I want to do is find the limit of p(t, x) as /\s -> 0. Actually, it will be that p(t, x) is a normal distribution in x, where t is the squareroot of the variance, and 0 is the mean. Any ideas on how to show this though by letting /\s -> 0? Thanks again for the help guys!

Edited May 8, 2010 by cosine
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[insane_alien]

well, brownian motion(in the physical world at least) isn't a continuous thing. its made up of discrete events(collisions) however the time between impacts and the distance travelled could be thought of as continuous random variables unless you want to bring quantum mechanics into this.

 

is usppose the best way to make it continuous would be to have the particle travel for a random time before there is a collision that adds a random momentum vector(random direction and magnitude) to it.

 

i don't think it is possible to have continuous collisions.

[the tree]

is usppose the best way to make it continuous would be to have the particle travel for a random time before there is a collision that adds a random momentum vector(random direction and magnitude) to it.

Would that really be continuous? Say a particle were moving horizontally for a while then there was a force applied that knocked it vertically so that it was travelling at a diagonal - wouldn't that point be a discontinuity?

 

i don't think it is possible to have continuous collisions.

Well if you had a force being applied all the time (not really a collision) that force could change in an unpredictable manner. If you're not looking at a literal particle but were instead trying to model a stock market then some changes wouldn't be discrete events.

[insane_alien]

well see, even in the stockmarket there are discrete events(company announcements, individual sales and purchases of stock etc) to treat either as continuous would be wrong to some extent.

 

i can't really think of an example where it would be a continuous movement as there is always a discrete event at some point. sure its possible to do it theoretically for a simulation but then you have to ask whether it accurately represents the system you are trying to simulate.

[the tree]

i can't really think of an example where it would be a continuous movement as there is always a discrete event at some point. sure its possible to do it theoretically for a simulation but then you have to ask whether it accurately represents the system you are trying to simulate.

Okay. Not specifically with motion but just in general: if you've got big numbers and long periods of time then it often makes sense to look at something as continuous even if it isn't really. Radioactive decay for instance - so long as the number of atoms involved is at least in the hundreds then exponential decay isn't a bad model. The "atto-fox" problem essentially just arises in boundary cases where a continuous model fails because the numbers are too small.

[insane_alien]

yes, re-reading the OP it isn't clear whether there is a large number of interactions or merely a few. he seems to be taking a constant timestep and having a single interaction at each time step.

 

it might help if he considers the bean machine (no, i'm not kidding http://en.wikipedia.org/wiki/Bean_machine). i think that might be closer to what he's looking for.

[cosine]

It can be intuitively difficult to imagine such a function because when we draw something it tends to be smooth (a.e. -for technicality's sake). But continuous Brownian Motion is an example of a function that is continuous everywhere but nowhere differentiable.

 

(Note: Brownian Motion is also referred to as a Weiner Process.)

The Wiener process plays an important role both in pure and applied mathematics. In pure mathematics, the Wiener process gave rise to the study of continuous time martingales. It is a key process in terms of which more complicated stochastic processes can be described. As such, it plays a vital role in stochastic calculus, diffusion processes and even potential theory. It is the driving process of Schramm-Loewner evolution. In applied mathematics, the Wiener process is used to represent the integral of a Gaussian white noise process, and so is useful as a model of noise in electronics engineering, instruments errors in filtering theory and unknown forces in control theory.

 

The Wiener process has applications throughout the mathematical sciences. In physics it is used to study Brownian motion, the diffusion of minute particles suspended in fluid, and other types of diffusion via the Fokker-Planck and Langevin equations. It also forms the basis for the rigorous path integral formulation of quantum mechanics (by the Feynman-Kac formula, a solution to the Schrödinger equation can be represented in terms of the Wiener process) and the study of eternal inflation in physical cosmology. It is also prominent in the mathematical theory of finance, in particular the Black–Scholes option pricing model.

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it might help if he considers the bean machine (no, i'm not kidding http://en.wikipedia.org/wiki/Bean_machine). i think that might be closer to what he's looking for.

 

hey insane_alien, I know what you mean by the bean machine, that is the sort of image I have in mind! Actually, what I want to see is that as the balls and spaces get smaller and more plentiful, the end result of the bean machine approximates a bell curve. I just want to see that in math language (aka by passing to the limit).

[timo]

The term is "Wiener process", not "Weiner process". I find the question interesting but sadly do not have (any!) time to think about it. There is a math statement that is called "central limit theorem", iirc. I am not sure if that really helps but it is what spontaneously comes to my mind.