Brownian Motion - Some Mathematical Properties

Some properties of Brownian motion

1. The increment Y(t+h) - Y(t) is normally distributed with mean 0 and standard deviation √h.

This is the normal increments property of Brownian motion.

2. If t1 < t2 < t3 < t4, then the increment Y(t2) - Y(t1) is independent of the increment Y(t4) - Y(t3).

This is the independent increments property of Brownian motion.

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,

Prob(Y(t + h) - Y(t) < u) = Prob(Y(s⋅(t + h)) - Y(s⋅t) <(√s)⋅u).

This is the self-affinity property of Brownian motion.

It can be interpreted as Y scales as √(time).

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.)

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