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);
5 \node[font=\small,align=center,font=\scriptsize] at (0.8,0.39) {0};
6 \draw[black!13,line width=.25pt] (2.975,0.65) -- (2.975,3.45);
7 \draw[] (2.975,0.59) -- (2.975,0.71);
8 \node[font=\small,align=center,font=\scriptsize] at (2.975,0.39) {0.5};
9 \draw[black!13,line width=.25pt] (5.15,0.65) -- (5.15,3.45);
10 \draw[] (5.15,0.59) -- (5.15,0.71);
11 \node[font=\small,align=center,font=\scriptsize] at (5.15,0.39) {1};
12 \draw[black!13,line width=.25pt] (7.325,0.65) -- (7.325,3.45);
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);
16 \draw[] (9.5,0.59) -- (9.5,0.71);
17 \node[font=\small,align=center,font=\scriptsize] at (9.5,0.39) {2};
18 \draw[black!13,line width=.25pt] (0.8,0.65) -- (9.5,0.65);
19 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (0.64,0.65) {0};
20 \draw[black!13,line width=.25pt] (0.8,1.316667) -- (9.5,1.316667);
21 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (0.64,1.316667) {0.25};
22 \draw[black!13,line width=.25pt] (0.8,1.983333) -- (9.5,1.983333);
23 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (0.64,1.983333) {0.5};
24 \draw[black!13,line width=.25pt] (0.8,2.65) -- (9.5,2.65);
25 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (0.64,2.65) {0.75};
26 \draw[black!13,line width=.25pt] (0.8,3.316667) -- (9.5,3.316667);
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 $\eta_1(s)$};
32 \draw[line width=1.1pt] plot coordinates {(0.8522,0.650001) (0.888383,0.651177) (0.924567,0.675089) (0.96075,0.772122) (0.996933,0.962153) (1.033116,1.221126) (1.0693,1.5109) (1.105483,1.799624) (1.141666,2.066984) (1.177849,2.302583) (1.214033,2.502789) (1.250216,2.668017) (1.286399,2.800825) (1.322582,2.904726) (1.358766,2.983489) (1.394949,3.040763) (1.431132,3.079892) (1.467315,3.103843) (1.503499,3.115192) (1.539682,3.116147) (1.575865,3.108582) (1.612049,3.094078) (1.648232,3.073964) (1.684415,3.04935) (1.720598,3.021164) (1.756782,2.99018) (1.792965,2.957042) (1.829148,2.922282) (1.865331,2.886345) (1.901515,2.849598) (1.937698,2.812344) (1.973881,2.774834) (2.010064,2.737272) (2.046248,2.699827) (2.082431,2.662637) (2.118614,2.625812) (2.154797,2.589441) (2.190981,2.553596) (2.227164,2.518332) (2.263347,2.483693) (2.299531,2.44971) (2.335714,2.416408) (2.371897,2.383802) (2.40808,2.351903) (2.444264,2.320715) (2.480447,2.290239) (2.51663,2.260472) (2.552813,2.23141) (2.588997,2.203042) (2.62518,2.175361) (2.661363,2.148355) (2.697546,2.122012) (2.73373,2.096317) (2.769913,2.071257) (2.806096,2.046819) (2.842279,2.022986) (2.878463,1.999744) (2.914646,1.977078) (2.950829,1.954972) (2.987013,1.933412) (3.023196,1.912384) (3.059379,1.891871) (3.095562,1.87186) (3.131746,1.852336) (3.167929,1.833286) (3.204112,1.814696) (3.240295,1.796552) (3.276479,1.778843) (3.312662,1.761554) (3.348845,1.744674) (3.385028,1.728191) (3.421212,1.712093) (3.457395,1.696369) (3.493578,1.681008) (3.529762,1.666) (3.565945,1.651334) (3.602128,1.637) (3.638311,1.622988) (3.674495,1.60929) (3.710678,1.595896) (3.746861,1.582798) (3.783044,1.569987) (3.819228,1.557454) (3.855411,1.545193) (3.891594,1.533195) (3.927777,1.521452) (3.963961,1.509959) (4.000144,1.498707) (4.036327,1.48769) (4.07251,1.476902) (4.108694,1.466336) (4.144877,1.455987) (4.18106,1.445847) (4.217244,1.435913) (4.253427,1.426177) (4.28961,1.416636) (4.325793,1.407283) (4.361977,1.398114) (4.39816,1.389123) (4.434343,1.380307) (4.470526,1.37166) (4.50671,1.363179) (4.542893,1.354859) (4.579076,1.346695) (4.615259,1.338685) (4.651443,1.330823) (4.687626,1.323107) (4.723809,1.315533) (4.759992,1.308096) (4.796176,1.300795) (4.832359,1.293625) (4.868542,1.286583) (4.904726,1.279666) (4.940909,1.272871) (4.977092,1.266196) (5.013275,1.259637) (5.049459,1.253191) (5.085642,1.246856) (5.121825,1.24063) (5.158008,1.234509) (5.194192,1.228492) (5.230375,1.222576) (5.266558,1.216758) (5.302741,1.211037) (5.338925,1.20541) (5.375108,1.199875) (5.411291,1.19443) (5.447474,1.189074) (5.483658,1.183803) (5.519841,1.178616) (5.556024,1.173512) (5.592208,1.168489) (5.628391,1.163544) (5.664574,1.158677) (5.700757,1.153884) (5.736941,1.149166) (5.773124,1.14452) (5.809307,1.139945) (5.84549,1.13544) (5.881674,1.131002) (5.917857,1.12663) (5.95404,1.122324) (5.990223,1.118082) (6.026407,1.113902) (6.06259,1.109783) (6.098773,1.105724) (6.134956,1.101724) (6.17114,1.097781) (6.207323,1.093895) (6.243506,1.090065) (6.27969,1.086288) (6.315873,1.082565) (6.352056,1.078894) (6.388239,1.075274) (6.424423,1.071705) (6.460606,1.068184) (6.496789,1.064712) (6.532972,1.061288) (6.569156,1.057909) (6.605339,1.054577) (6.641522,1.051289) (6.677705,1.048045) (6.713889,1.044844) (6.750072,1.041686) (6.786255,1.038569) (6.822438,1.035493) (6.858622,1.032457) (6.894805,1.02946) (6.930988,1.026502) (6.967172,1.023582) (7.003355,1.0207) (7.039538,1.017854) (7.075721,1.015044) (7.111905,1.012269) (7.148088,1.009529) (7.184271,1.006823) (7.220454,1.004151) (7.256638,1.001511) (7.292821,0.998904) (7.329004,0.996329) (7.365187,0.993786) (7.401371,0.991273) (7.437554,0.98879) (7.473737,0.986337) (7.509921,0.983914) (7.546104,0.981519) (7.582287,0.979152) (7.61847,0.976814) (7.654654,0.974502) (7.690837,0.972218) (7.72702,0.969961) (7.763203,0.967729) (7.799387,0.965523) (7.83557,0.963342) (7.871753,0.961187) (7.907936,0.959055) (7.94412,0.956948) (7.980303,0.954865) (8.016486,0.952804) (8.052669,0.950767) (8.088853,0.948753) (8.125036,0.946761) (8.161219,0.94479) (8.197403,0.942842) (8.233586,0.940914) (8.269769,0.939008) (8.305952,0.937122) (8.342136,0.935256) (8.378319,0.933411) (8.414502,0.931585) (8.450685,0.929779) (8.486869,0.927992) (8.523052,0.926223) (8.559235,0.924474) (8.595418,0.922742) (8.631602,0.921029) (8.667785,0.919333) (8.703968,0.917655) (8.740151,0.915994) (8.776335,0.914351) (8.812518,0.912724) (8.848701,0.911113) (8.884885,0.909519) (8.921068,0.907941) (8.957251,0.906379) (8.993434,0.904833) (9.029618,0.903302) (9.065801,0.901786) (9.101984,0.900285) (9.138167,0.898799) (9.174351,0.897328) (9.210534,0.895871) (9.246717,0.894428) (9.2829,0.892999) (9.319084,0.891584) (9.355267,0.890183) (9.39145,0.888795) (9.427633,0.88742) (9.463817,0.886059) (9.5,0.88471)};
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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