Re: brownian motion

On 15 May 2006 17:36:42 -0700, "Fedor" <malabar_carotte@xxxxxxxxx>
wrote:

Hi all,

I have a question about brownian motion in R^n. Suppose that B_t(w)
is a brownian motion that starts from the origine and T(w) is the first
time that t->B_t(w) touches the unit sphere. What is the law of T ? I
can only do this for n=1 but I can't generalize for higher dimension
:-(

It's a unformly distributed measure on the sphere (ie
rotation-invariant). This is clear because brownian
motion is rotation-invariant, in a suitable sense.
(Which in turn is clear because a gaussian distribution
is rotation-invariant... or at least the gaussian
distributions that come up here.)

This fact is the essential reason why you can use
brownian motion to solve the Dirichlet problem
(from comments elsewhere I gather you know what
I mean by that). And then the fact that brownian
motion gives a solution to the Dirichlet problem
shows that if you start brownian motion at a point
of a ball other than the center and look at the
first point it hits the boundary the distribution
of that point is given by the Poisson kernel.

Thank you,
Fedor.


************************

David C. Ullrich
.

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