Brownian Motion - Brownian Bridges

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 0 ≤ t ≤ T, and X(0) = Y(0) = 0.
Then the path (X(t) - (t/T)⋅X(T), Y(t) - (t/T)⋅Y(T)) begins and ends at the origin, and is called a Brownian Bridge.
Visual comparisons between Brownian cluster simulations and the coastlines of islands led Benoit Mandelbrot to conjecture that the periphery of a Brownian Bridge (the part that can be reached from far away without crossing any other point of the cluster) has dimension 4/3.
Recently this was proved by Werner, Schramm, and Lawler.

Return to Brownian Motion.