# Brownian Motion on Nested Fractals by Tom Lindstrom By Tom Lindstrom

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12 and t h e easy t o check that fact Y that (p , . . , p h a s t h e same ) is distribution is a collection of n-points, this means that has the same hitting distribution as both ^ hence the proposition is proved. , , and n-f 1 Y VI. Transition times So far our processes are just Markov chains moving one B r n step at each unit time. To construct a limit process, we must know how to rescale time as n grows large, and a convenient way of formulating this problem is in terms of random transition times as explained informally in Chapter III.

Thus U=U |x-y|= |x"-y | , „° XX" and hence 33 U M leaves y U . does the job. yy» If we let 60=min{|x-y|: x,y GF,x*y}, then the shortest distance between two elements in the same n-cell is 6 =v 60. Two distinct points in E are n-neighbors if they are n-points belonging to the same n-cell; they are nearest n-neighbors if in addition the distance between them is 6 . ,s such that s. ,, are n-neighbors for all i

Is a walk which connects r r-1 2 y. Hence the image of this walk under which connects U x and y without hitting x* and z and avoids is a strict O-walk z. 11 Proposition. ,s 1 p such that . Then there is a strict 1-walk x=s , y=s 1 p and s \$F for all i, l