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The article makes no sense. Einstein realized that matter was composed out of atoms, so the number of collisions of a Brownian particle with the surrounding molecule is finite in a finite period of time. However, for times $t$ much longer than the typical scale between the collisions, the particle moves by a distance scaling like $\sqrt{t}$. It follows that ...

There are tons of papers on the connection between quantum processes and probability theory (though I don't understand why you single out coherent states - they don't play a special role in this connection). The theory of stochastic processes and the theory of quantum processes are the commutative and noncommutative side of the same coin, with many ...

The differential is used to specify that the number is for a "differential range", which is a way to remind you that the notions involved are somewhat fuzzy. Let me give a purely mathematical example. Suppose I tell you that I am going to pick an arbitrary real number between 0 and 10, with the likely hood of a number being picked being proportional to the ...

For Brownian motion, Langevin equation, Fokker-Planck equations, Stochastic process.. from the viewpoint of physicists, the following are standard references: Brownian Motion: Fluctuations, Dynamics, and Applications The Fokker-Planck Equation: Methods of Solutions and Applications Handbook of Stochastic Methods: for Physics, Chemistry and the Natural ...

Many different types of connection can be made between stochastic states over commutative algebras of observables and quantum states over noncommutative algebras of observables. As Arnold says, there is a substantial literature. One approach is to construct both classical and quantum models in a formalism that accommodates both; within the structured ...

The equation resembles the heat equation very closely, so much so, that it is the heat equation. The literature on elementary things like this is uniformly terrible, so I cannot give a reference. To see why it is the heat equation, note that the law for the distribution function is the same as for a probability distribution, so it is linear equation, by the ...

This is a bit strange. The Langevin equation $$ \frac{dv}{dt}~+~\beta v~=~\frac{F}{m} $$ for the motion of a free particle under a stochastic force $F$ evaluates the velocity as an average or in an interval. The stochastic force has a Gaussian probability distribution $\langle F(t)F(t’)\rangle~=$ $2\beta kT\delta(t~-~t’)$, which is also a Markov ...