In 3D a random walk taking steps of size 1 with independent and uniform angles is not dense. So let rho be such a path. Can I find a ball B (not circle) which rho does not interesect, and reflect rho inside ball, and then find a fractal dimension 2 for rho at the center of B, where the fractal dimension is defined as above?

Hi Douglas, Interesting point. My intuition was that when you smooth the Brownian motion, and you remove its fine detail at short length scales, then you reduce its Hausdorff dimension to 1. Can a curve with Hausdorff dimension 1 fill the plane? Thanks for the comment.

For a fractal dimension at the center of circle C, I suggest: Let T be an interval of time t. Let rho(T) denote the path of rho(t) evaluated on T. Let ^rho(T) denote rho(T) after it has been smoothed and inverted inside of circle C. Let c denote the center of C. Consider a circle Gamma(R) of radius R also centered on c, where R is smaller than the radius of C. Measure the length of ^rho(T) that is contained in Gamma(R), and denote this by L(R). I claim that in the limit as the interval T becomes large, and R becomes small, L(Ralpha) = alpha^DL(R), where this defines fractal dimension D.

To smooth it out, start with a time t_o and a point r_o = rho(t_o). Find the maximum time t_1 such that there exists a circle of radius R that contains rho(t) for all t in the interval t_o < t < t_1. So now we have a second point r_1 = rho(t_1) on rho. These two points are a distance R apart. Then repeat: Find the maximum time t_2 such that there exists a circle of Radius R that contains rho(t) for all t in the interval t_1 < t < t_2. Repeat for all positive integer i. We can "turn" this procedure "around" to negative integers. Then these points can be joined with a continuous spline.

Hi Shawn, Do you have a succinct turn of phrase to describe, "the maximum value of the radial part for t < T, as a function of T"?. I need something I can say instead of "First Passage Time". I know this is a picky point, but it would help me write my paper. Thanks.

Here is why I thought they might be equivalent: If you know that the expected first passage time to a circle of radius Ro is To, then conversely you know that, at time To, the expected maximum radial part of the motion for t<To is Ro. Is there a flaw this logic? I think one could flesh this out into a full argument that if you know the first passage time as a function of R, then you can figure out the maximum radial component for t < T as a function of T. Thanks for your time. Chris

Yes, you could call in "the expected value of the maximum value of the radial part of the Brownian motion for t<T." Is there a succinct way of referring to this kind of problem? Would you agree this is equivalent to a first passage time problem.