Infinitesimal Calculus of Random Walk and Poisson Processes
Infinitesimal Calculus of Random Walk and Poisson Processes
Infinitesimal Calculus of Random Walk and Poisson Processes
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Gauge Institute Journal,<br />
H. Vic Dannon<br />
<strong>Infinitesimal</strong> <strong>Calculus</strong> is the <strong>Calculus</strong> <strong>of</strong> hyper-real<br />
functions.<br />
<strong>Infinitesimal</strong> <strong>Calculus</strong> is far more effective than the<br />
εδ ,<br />
<strong>Calculus</strong>, because being based on almost zero numbers, it<br />
allows us to deal with their reciprocals, the almost infinite<br />
numbers. We have no use for infinity by itself, but to<br />
comprehend the effects <strong>of</strong> singularities, we have use for the<br />
almost infinite.<br />
<strong>Infinitesimal</strong>s are a precise tool compared to the vague limit<br />
concept, <strong>and</strong> the awkward<br />
εδ ,<br />
statements.<br />
<strong>R<strong>and</strong>om</strong> walks are made clearer with infinitesimals.<br />
<strong>Poisson</strong> Process can be derived only in <strong>Infinitesimal</strong><br />
<strong>Calculus</strong>.<br />
0.2 <strong>R<strong>and</strong>om</strong> <strong>Processes</strong><br />
Probability Distributions are defined on <strong>R<strong>and</strong>om</strong> Variables.<br />
<strong>R<strong>and</strong>om</strong> Variables assign numerical vales to outcomes.<br />
Thus, maps outcomes into the real line.<br />
<strong>R<strong>and</strong>om</strong> Variables that evolve in time are called <strong>R<strong>and</strong>om</strong><br />
<strong>Processes</strong>, in Mechanics, or <strong>R<strong>and</strong>om</strong> Signals, in Electricity.<br />
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