3
$\begingroup$

Hi

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.

Thanks.

$\endgroup$
2
  • 1
    $\begingroup$ 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
5
$\begingroup$

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).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.