Chapter 4 - Equilibrium State

One of the powerful feature of Brownian motion is self similarity properties, extended from the Central Limit Theorem. The density function of $X$, $dX$, and $dW$ are all normal distribution. It is pure and simple form of mathematical expression, but it has some issues with reality.

For example, when observing particles moving in space as the Brownian motion simulation (from Wikipedia), in a very short time interval, do we really see a normal distribution, or rather a deterministic movement - a Dirac Delta jump function?

In the proposed process, the density function of $X$ and $d\xi$ are both Laplace distribution, but the $dX$ will vary depending on the time interval (still a combination of Laplace distribution with different variance) as below:

Assuming $p$ is the probability of jump occur within any time $\Delta t$, the following equation holds true for Laplace random variable $L$:
$$L(\sigma )=[p\delta +\mathrm{(1-}p)L(\sigma )]*L(\sigma \sqrt{p})$$ The equation is a little bit confusing. But it can be broken down into two parts (a) with in the brackets on the right, it is the density function of "jump to coordinate origin 0", and then it's convoluted with (b) another Laplace distribution with variance $\sigma \sqrt{p}$. The convolution is exactly how it looked like right before the observation. In other words, in the equilibrium state, the particle ball maintains a finite entropy with Laplace distribution.

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