Paper VI: Transfer Semigroups: Stochastic, Bayesian, Fractional, and PDE Realizations

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2 \path[use as bounding box] (0,-.6) rectangle (14,4.8);
3 \node[font=\small,align=center,font=\small] at (2.17,4.45) {$8K$};
4 \path[draw=black!30,fill=black!8] (0.65,2.99) rectangle (1.41,3.75);
5 \node[font=\small,align=center,font=\small] at (1.03,3.37) {$5$};
6 \path[draw=black!30,fill=black!8] (1.41,2.99) rectangle (2.17,3.75);
7 \node[font=\small,align=center,font=\small] at (1.79,3.37) {$1$};
8 \path[draw=black!30,fill=black!8] (2.17,2.99) rectangle (2.93,3.75);
9 \node[font=\small,align=center,font=\small] at (2.55,3.37) {$2$};
10 \path[draw=black!30,fill=white] (2.93,2.99) rectangle (3.69,3.75);
11 \node[font=\small,align=center,font=\small] at (3.31,3.37) {$0$};
12 \path[draw=black!30,fill=black!8] (0.65,2.23) rectangle (1.41,2.99);
13 \node[font=\small,align=center,font=\small] at (1.03,2.61) {$1$};
14 \path[draw=black!30,fill=black!8] (1.41,2.23) rectangle (2.17,2.99);
15 \node[font=\small,align=center,font=\small] at (1.79,2.61) {$5$};
16 \path[draw=black!30,fill=white] (2.17,2.23) rectangle (2.93,2.99);
17 \node[font=\small,align=center,font=\small] at (2.55,2.61) {$0$};
18 \path[draw=black!30,fill=black!8] (2.93,2.23) rectangle (3.69,2.99);
19 \node[font=\small,align=center,font=\small] at (3.31,2.61) {$2$};
20 \path[draw=black!30,fill=black!8] (0.65,1.47) rectangle (1.41,2.23);
21 \node[font=\small,align=center,font=\small] at (1.03,1.85) {$2$};
22 \path[draw=black!30,fill=white] (1.41,1.47) rectangle (2.17,2.23);
23 \node[font=\small,align=center,font=\small] at (1.79,1.85) {$0$};
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31 \node[font=\small,align=center,font=\small] at (1.79,1.09) {$2$};
32 \path[draw=black!30,fill=black!8] (2.17,0.71) rectangle (2.93,1.47);
33 \node[font=\small,align=center,font=\small] at (2.55,1.09) {$1$};
34 \path[draw=black!30,fill=black!8] (2.93,0.71) rectangle (3.69,1.47);
35 \node[font=\small,align=center,font=\small] at (3.31,1.09) {$5$};
36 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (0.5,3.37) {0};
37 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (0.5,2.61) {1};
38 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (0.5,1.85) {2};
39 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (0.5,1.09) {3};
40 \node[font=\small,align=center,font=\scriptsize] at (1.03,3.98) {0};
41 \node[font=\small,align=center,font=\scriptsize] at (1.79,3.98) {1};
42 \node[font=\small,align=center,font=\scriptsize] at (2.55,3.98) {2};
43 \node[font=\small,align=center,font=\scriptsize] at (3.31,3.98) {3};
44 \node[font=\small,align=center,font=\small] at (7.56,4.35) {$CK$: aggregate transition columns};
45 \path[draw=black!30,fill=black!8] (5.4,2.57) rectangle (6.48,3.65);
46 \node[font=\small,align=center,font=\small] at (5.94,3.11) {$\frac{3}{4}$};
47 \path[draw=black!30,fill=black!8] (6.48,2.57) rectangle (7.56,3.65);
48 \node[font=\small,align=center,font=\small] at (7.02,3.11) {$\frac{3}{4}$};
49 \path[draw=black!30,fill=black!8] (7.56,2.57) rectangle (8.64,3.65);
50 \node[font=\small,align=center,font=\small] at (8.1,3.11) {$\frac{1}{4}$};
51 \path[draw=black!30,fill=black!8] (8.64,2.57) rectangle (9.72,3.65);
52 \node[font=\small,align=center,font=\small] at (9.18,3.11) {$\frac{1}{4}$};
53 \path[draw=black!30,fill=black!8] (5.4,1.49) rectangle (6.48,2.57);
54 \node[font=\small,align=center,font=\small] at (5.94,2.03) {$\frac{1}{4}$};
55 \path[draw=black!30,fill=black!8] (6.48,1.49) rectangle (7.56,2.57);
56 \node[font=\small,align=center,font=\small] at (7.02,2.03) {$\frac{1}{4}$};
57 \path[draw=black!30,fill=black!8] (7.56,1.49) rectangle (8.64,2.57);
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59 \path[draw=black!30,fill=black!8] (8.64,1.49) rectangle (9.72,2.57);
60 \node[font=\small,align=center,font=\small] at (9.18,2.03) {$\frac{3}{4}$};
61 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (5.25,3.11) {A};
62 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (5.25,2.03) {B};
63 \node[font=\small,align=center,font=\scriptsize] at (5.94,3.88) {0};
64 \node[font=\small,align=center,font=\scriptsize] at (7.02,3.88) {1};
65 \node[font=\small,align=center,font=\scriptsize] at (8.1,3.88) {2};
66 \node[font=\small,align=center,font=\scriptsize] at (9.18,3.88) {3};
67 \node[font=\small,align=center,font=\small] at (7.55,0.6) {Columns 0 and 1 agree; columns 2 and 3 agree.};
68 \node[font=\small,align=center] at (12,3.55) {$\bar K=\begin{pmatrix}3/4&1/4\\1/4&3/4\end{pmatrix}$};
69 \node[font=\small,align=center,text width=2.5cm,font=\small] at (12,1.6) {Fine trajectories differ.\\Coarse predictions agree.};
70 \end{tikzpicture}

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