Diffusion

by Nikolai V. Shokhirev

Up: Stochastic Labs
Pseudorandom numbers
-  Probability transformation
Diffusion, Wiener process
Geometric Brownian motion
-  Ornstein-Uhlenbeck process

Wiener process Wt

This is a one-dimensional (say, x) continuous process. For any two times  t > s it has independent increments Wt - Ws distributed Normally with mean 0 and variance (t - s).

Stochastic Differential Equation (SDE)

The above verbal definition can be presented in the following compact form

dx = dWt

Consider this a s a formal definition because x(t) is not differentiable and cannot be governed by any differential equation.

Diffusion with drift

The simplest generalization of the Wiener process is the diffusion with drift where mean is linear in time with some factor μ and its variance is σ2 (t - s).

SDE

dx = μ t + σ dWt

Fokker-Planck Equation

The above SDE and the corresponding solution techniques (Itô calculus or Stratonovich calculus) are the ways to calculate mean (observable) values. The alternative way is the solution of the following partial differential (PDE) equation for the probability density function ρ(x, t):

This equation is also known as Kolmogorov forward equation. The solution for ρ(x, 0; x0) = δ(x - x0)  is

Here t0 = 0.

Stochastic simulation program

Info

Diffusion

Feel free to download and play (at your own risk). I am adding several new features, please check later.

Downloads

Stochastic simulation program.

Some references

ABC Tutorials | Data Processing | Indirect Measurements | NMR Tutorials

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