Markov diffusion process
Diffusion processes are not deterministic but continous. It is derived by assuming a transition $W$ like in the Markov jump process, that only depends on the jump increment $y = x - x^\prime$: $W(x|x^\prime,t) = f(x^\prime, y, t)$.
The Chapman-Kolmogorov equation becomes the Fokker-Planck equation $$\frac{\partial}{\partial p}p(x,t) = -\frac{\partial}{\partial x_i}[g_i(x,t)p(x,t)] + \frac{1}{2}\frac{\partial^2}{\partial x_i \partial x_j} [D_{ij}(x,t)p(x,t)],$$ where same indices are summed over and $$g_i(x,t)=\int dy , y_i f(x,y,t), \quad D_{ij}=\int dy ,y_iy_jf(x,y,t).$$ Same equation hold for the Propagator
- First term on the right is the deterministic part that develops according to $g$
- Second term is the diffusion of $X(t)$
# Brownian motion
For $g = 0$ and $D \equiv 1$ the propagator becomes a Gaussian. This is called the Borwnian motion process
With the initial condition $p(x,t=0) = \delta(x)$ this becomes the Wiener process