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I have read Stochastic Differential Equations by Bernt Oksendal
It constructs Brownian motion by Kolmogorov extension theorem by consider $p(t,x,y)=(2\pi t)^{-n/2} e^{- \frac{|x-y|^{2}}{2t}}$
But I can't understand what is the relation to the Brownian motion in physics.

user4933
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  • I found E. Nelson: "Dynamical Theories of Brownian Motion", https://web.math.princeton.edu/~nelson/books/bmotion.pdf nice to read. It discusses the history and the connection of Wiener processes with more realistic stochastic particle dynamics. –  Jun 09 '14 at 09:02
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    It kindoff gives a proof for the existence of atoms, is this the kind of thing you are looking for ? – Nick Jun 09 '14 at 09:36

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