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| Some early history of Brownian motion. |
| Brownian motion in the plane, and a first look at scaling. |
| A clearer picture is the graph of Brownian motion in 1 dimension. |
| Here are some of the mathematical properties of Brownian motion. |
| For comparison with data, here are the visual signatures of Brownian motion. Anticipating later work, for amusement we compare these graphs with some stock market data. |
| Finally, we mention the Brownian bridge, a related construction of some theoretical importance now. |
Pictured here on the left is a portion of the trail left by a particle undergoing Brownian motion in the plane. To emphasize the scaling of Brownian trails, the first quarter of the trail is contained in the box on the left. This box is magnified by a factor of 2 (horizontally and vertically) to give the box on the right, and the first quarter of the left trail is rescaled within this box. (Later parts of the first trail reenter this box, so some of the parts shown on the left box are not shown in the right box.) More detail is revealed in the magnification, and the left side and right side look very similar.
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Brownian motion is easier to understand in the one-dimensional case. Suppose a particle (red) moves only along the y-axis. We can graph (green) its motion as a function of time (t on the horizontal axis). Click the picture for an animation.
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To understand the properties of 1-dimensional Brownian motion, here is the graph of a longer simulation, 2000 points. We call this the graph of the function Y(t).
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Properties of Brownian motion:
1. The increment Y(t+h) - Y(t) is normally distributed with mean 0 and standard deviation sqrt(h). We call this the normal increments property of Brownian motion.
2. If t1 < t2 < t3 < t4,
then the increment
3. For all h > 0, the increment Y(t + h) - Y(t) is independent of t. That is, Brownian motion is stationary.
4. For any number u and any numbers s,t > 0,
5. With probability 1, Y(t) is continuous and Y(0) = 0.
6. A Brownian path in n-dimensional space, n > 1, has dimension = 2.
7. The graph (Y vs t) of one-dimensional Brownian motion has dimension 3/2.
8. A Brownian path in the plane has double points, triple points, quadruple points, and multiple points of all orders.
9. A Brownian path in 3-dimensional space has double points but no triple points. (With probability 1, a smooth curve in 3-dimensional space has no double points.)
We shall see that most real data do not share these properties. The comparisons are clearest when we view differences instead of the raw data. Here are difference graphs for two examples of Brownian motion. For comparison, below these are difference graphs for four years of EMC closing prices (left) and Dell closing prices (right).
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For later reference we note that when the thickness of the plotting line is approximately equal to the spacing between successive values, the bulk of the difference plot merges into a band. Further, for Brownian motion (top two graphs) we observe
1. The band is of approximately constant width.
2. The outliers are relatively small (this is called the short tails property).
3. The outliers are approximately uniformly distributed.
How well do real data satisfy these conditions? At least for EMC and Dell, the match appears to be weak.
Finally, we mention a recent theoretical development. A Brownian path in two dimensions can be
produced from two independent one-dimensional Brownian functions X(t) and Y(t). Suppose we consider
the range
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