I know about the Brownian bridge, for example $B_t = W_t - tW(1)$. Is it possible to create it in 2D? ie, to have a 2D Brownian motion, which constitutes a surface, and have it return to 0 when the distance (according to some metric) from the center is equal to some constant?

For example, to have an image which is a Brownian motion realization and have it conditioned as equal to 0 on the unit circle.


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    $\begingroup$ What exactly do you mean by "2D Brownian motion, which constitutes a surface"? $\endgroup$ – j.c. Aug 4 '12 at 16:05
  • $\begingroup$ I wrote it to differentiate it from a 2D BM which has BM in x and y axis which gives us a "line". $\endgroup$ – id0 Aug 5 '12 at 10:57

A natural generalization of the Brownian bridge that should be readily adaptable to your problem is furnished by a Gaussian free field (see in particular the picture on the wiki page).


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