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 we’ve 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
realvalued 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 ChapmanKolmogorov equation, and hence that there exist
Markov
processes with the desired finitedimensional distributions.
60
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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
π
(
t
2

t
1
)
e

u
2
2(
t
2

t
1
)
(11.2)
=
B
dw
2
1
2
π
(
t
2

t
1
)
e

(
w
2

w
1
)
2
2(
t
2

t
1
)
(11.3)
≡
μ
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
times
t
1
≤
t
2
≤
t
3
,
μ
t
1
,t
2
μ
t
2
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 Spring '06
 Schalizi
 Normal Distribution, Logistic map, markov examples

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