By Tom Lindstrom
ISBN-10: 0821824848
ISBN-13: 9780821824849
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57 BROWNIAN MOTION ON NESTED FRACTALS Nonstandard measure theory provides a straight-forward proof of the following crucial lemma. For the necessary background information, see Chapter 3 of [l ]. 4 Lemma. (T ,d ) is complete. Proof: If {x } ^ is a Cauchy-sequence, let n n£ tsf nonstandard version. Each ~k a d d i t i v e measure on A T(A)=L(TN) st x n is an internal, *-countably 2 ~k R . Choose an i n f i n i t e e l e m e n t N£ IN, + A be the Loeb-measure of xXT, and define a measure x N let L(x^T) N on R by where — A {x J1 * fcT be its n n ^ IN (st (A)), is the standard part map.
This map provides a one-to-one correspondence between paths of equal probability. Given an equivalence class (x,y)£c. of ~, choose a pair and define By the lemma, call c. ,p p is independent of the choice of (x,y). I shall composite transition probabilities. My aim in this section is to show that the map P(P 1 '•••/P r )=(p ] ,••-,P r ) has a fixed point, and to do so I need to know that continuous and maps 9 to p is 9. 2 Proposition. If (p , . . , then (p.. , . . ,p )fc/?. Proof: For the purpose of this proof, it's convenient to modify the construction above slightly; instead of working with the Markov chain B , I shall use a chain probabilities hqv if **y if x=y.
Are n-neighbors for all iBrownian Motion on Nested Fractals by Tom Lindstrom
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