random_clock.tex
Paper VI: Transfer Semigroups: Stochastic, Bayesian, Fractional, and PDE Realizations
random_clock.tex
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1 \begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]
2 \path[use as bounding box] (0,-.6) rectangle (14,4.8);
3 \draw[black!13,line width=.25pt] (0.8,0.65) -- (0.8,3.45);
4 \draw[] (0.8,0.59) -- (0.8,0.71);
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6 \draw[black!13,line width=.25pt] (2.975,0.65) -- (2.975,3.45);
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8 \node[font=\small,align=center,font=\scriptsize] at (2.975,0.39) {0.5};
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10 \draw[] (5.15,0.59) -- (5.15,0.71);
11 \node[font=\small,align=center,font=\scriptsize] at (5.15,0.39) {1};
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13 \draw[] (7.325,0.59) -- (7.325,0.71);
14 \node[font=\small,align=center,font=\scriptsize] at (7.325,0.39) {1.5};
15 \draw[black!13,line width=.25pt] (9.5,0.65) -- (9.5,3.45);
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25 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (0.64,2.65) {0.75};
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27 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (0.64,3.316667) {1};
28 \draw[->,line width=.6pt] (0.8,0.65) -- (9.62,0.65);
29 \draw[->,line width=.6pt] (0.8,0.65) -- (0.8,3.57);
30 \node[font=\small,align=center,font=\scriptsize] at (5.15,0.1) {diffusion time $s$};
31 \node[font=\small,align=center,anchor=west,font=\scriptsize] at (0.8,3.85) {clock density $\eta_1(s)$};
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33 \draw[densely dotted] (1.525,0.65) -- (1.525,3.116885);
34 \node[font=\small,align=center,font=\scriptsize] at (3,4.45) {Clock mode at $s=1/6$};
35 \node[font=\small,align=center] at (11.8,3.8) {$\displaystyle\int_0^\infty\eta_t(s)\,ds=1$};
36 \node[font=\small,align=center] at (11.8,2.7) {$\displaystyle\int_0^\infty s\eta_t(s)\,ds=\infty$};
37 \node[font=\small,align=center,font=\small] at (11.8,1.2) {A probability clock\\need not have a finite mean.};
38 \end{tikzpicture}
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