These lecture notes were written from studying the material provided by Sungsoo Ahn on their personal website. To read this source material please click here.
1 Fokker-Planck Equation
1.1 Definition & Intuition
Consider a \(d\)-dimensional stochastic process \((X_t)_{t\in[0,T]}\) where \(X_t\in\mathbb{R}^d\) with dynamics given by the SDE
\[ \begin{cases} dX_t = f(X_t, t)dt + g(t) dW_t \\ X_0 \sim p_0 \end{cases} \]
where:
- \(f:\mathbb{R}^d\times [0,T]\to \mathbb{R}^d\) is the drift term.
- \(g:[0,T]\to \mathbb{R}\) is the diffusion term.
- \(W_t\) is a \(d\)-dimensional Wiener process.
- \(p_0\) is the initial distribution of \(X_0\).
The Fokker-Planck Equation (FPE) is the PDE for the marginal density \(p_t(x)\) of \(X_t\) at time \(t\). Intuitively, the SDE describes the evolution of a single particle and the FPE describes the probability of finding the particle at position \(x\) at time \(t\).
The Fokker-Planck Equation (FPE) is a PDE that describes the time evolution of \(p_t(x)\)
\[ \frac{\partial p_t(x)}{\partial t} = \underbrace{-\nabla \cdot (f(x,t)p_t(x))}_{\text{drift advection}} + \underbrace{\frac{g^2(t)}{2}\Delta_x p_t(x)}_{\text{diffusion spread}}, \]
where \(\nabla_x\) is the gradient (divergence) operator and \(\Delta_x = \sum_{i=1}^D \frac{\partial^2}{\partial x_i^2}\) is the Laplacian operator with respect to \(x\).
1.2 Drift Advection
The quantity \(fp_t\) is the probability flux, that is the evolution of the probability density due to the drift term. The negative Divergence of the probability flux measures \(-\nabla\cdot (fp_t)\) measures the net inflow of probability to a small region around \(x\).
For \(d=1\) this becomes the finite difference approximation
\[ \partial_t p \approx -\frac{(f\cdot p)(x+\Delta x)- (f\cdot p)(x)}{\Delta x}. \]
1.3 Diffusion Spread
At each instant, the SDE’s noise perturbs the particle by a symmetric random displacement \(gdW\). The aggregate effect is Gaussian blurring: peaks erode and valleys fill in.