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Unformatted text preview: Chapter 11 Markov Examples Section 11.1 finds the transition kernels for the Wiener process, as an example of how to manipulate such things. Section 11.2 looks at the evolution of densities under the action of the logistic map; this shows how deterministic dynamical systems can be brought under the sway of the theory weve developed for Markov processes. 11.1 Transition Kernels for the Wiener Process We have previously defined the Wiener process (Examples 38 and 78) as the real-valued process on R + with the following properties: 1. W (0) = 0; 2. For any three times t 1 t 2 t 3 , W ( t 3 )- W ( t 2 ) | = W ( t 2 )- W ( t 1 ) (inde- pendent increments); 3. For any two times t 1 t 2 , W ( t 2 )- W ( t 1 ) N (0 , t 2- t 1 ) (Gaussian increments); 4. Continuous sample paths (in the sense of Definition 72). Here we will use the Gaussian increment property to construct a transition kernel, and then use the independent increment property to show that these keernels satisfy the Chapman-Kolmogorov equation, and hence that there exist Markov processes with the desired finite-dimensional distributions. 60 CHAPTER 11. MARKOV EXAMPLES 61 First, notice that the Gaussian increments property gives us the transition probabilities: P ( W ( t 2 ) B | W ( t 1 ) = w 1 ) = P ( W ( t 2 )- W ( t 1 ) B- w 1 ) (11.1) = B- w 1 du 1 2 ( t 2- t 1 ) e- u 2 2( t 2- t 1 ) (11.2) = B dw 2 1 2 ( t 2- t 1 ) e- ( w 2- w 1 ) 2 2( t 2- t 1 ) (11.3) t 1 ,t 2 ( w 1 , B ) (11.4) To show that W ( t ) is a Markov process, we must show that, for any three...
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This note was uploaded on 12/20/2011 for the course STAT 36-754 taught by Professor Schalizi during the Spring '06 term at University of Michigan.
- Spring '06