Pure Mathematics

Brownian Motion on Nested Fractals by Tom Lindstrom

Posted On March 23, 2017 at 8:01 pm by / Comments Off on Brownian Motion on Nested Fractals by Tom Lindstrom

By Tom Lindstrom

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Extra info for Brownian Motion on Nested Fractals

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X p,2'•••' 1 p,N and by 0 0 above y=<|>. x F. 1 i,i' — ' F. V i p. is an n-point, the number of N-complexes belongs to is the same for all N>n. Proof: Assume that E. x l ,1 ' #1 are the n-complexes X=(|>. for e a c h x <\>. in ,E. 1 m,T 'Vn belongs to, and let ° j n, x belongs to m N-complexes E. l Assume that x . 1l . ^ -1,-'^ j

I shall describe this process by specifying its transition probabilities q X/ y . If x and y are not n-neighbors belong to the same n-cell, or x = y ) , then are n-neighbors, there exist a sequence distinct points x' ,y' £F such that y=<\>. °c|;. (y ' ) . The pair 1 1 n class q = 0 . ,i x and y and two x= cj>. © . . •4». e. e. the number of n-cells belongs to. 16 the multiplicity is independent of n. A Markov chain probabilities q B with transition n is called a Markov chain induced by (p , .

Since B z z€V, be the the 38 TOM LINDSTR0M intersection B flV must h a v e p o s i t i v e Vol (B flV) z V o l (B ) volume, and hence > j^ K z for some positive integer belongs to and let If B z. K. Choose N K N-cells . ° °. (F), D 3 1,l 1,N be the pre-image of x so large that , 4>. ° D K,1 x€F °4>. (F), D K,N under the i-th of these maps is the closed ball with center x -N v , the and radius family U. ° ° ck 3 ^i,l i,N (B nv)}. Z 1