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. |
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2. If t1 < t2 < t3 < t4,
then the increment |
This is the independent increments property of Brownian motion. |
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3. For all h > 0, the increment Y(t + h) - Y(t) is independent of t. |
That is, Brownian motion is stationary. |
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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). |
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5. With probability 1, Y(t) is continuous and Y(0) = 0. |
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6. A Brownian path in n-dimensional space, n > 1, has dimension = 2. |
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7. The graph (Y vs t) of one-dimensional Brownian motion has dimension 3/2. |
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8. A Brownian path in the plane has double points, triple points, quadruple points, and multiple points of all orders. |
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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.) |
Return to Brownian motion.