>[!quote] In a Nutshell >Partial differential equation that describes the **deterministic** evolution of a [[Probability Distribution|probability distribution]] over time under the effect of drift forces and random forces / noise. >[!brainwaves] Idea >For [[SDE - Stochastic Differential Equation|SDEs]], we cannot make prediction about single points due to the randomness. However, we can focus on the evolution of the [[Probability Density Function|probability density]] instead, which is governed by [[PDEs - Partial Differential Equations|PDEs]]. --- #### One Dimension Assume an [[Itô Diffusion|Itô process]] driven by a [[Wiener Process or Brownian Motion|Wiener process]] $W_t$ with diffusion coefficient $D(x,t)=\sigma^{2}(X_t,t)/2$ and drift term $\mu(X_t, t)$ of the form $dX_t=\mu(X_t, t)dt+\sigma(X_t,t)dW_t.$The Fokker-Planck equation for the [[Probability Density Function|probability density]] $p(x,t)$ of the [[Random Variable|random variable]] $X_t$ is $\frac{\partial p(x,t)}{\partial t}=-\frac{\partial}{\partial x}\Big[ \mu(x,t) p(x,t) \Big]+\frac{\partial^{2}}{\partial x^{2}}[D(x,t)p(x,t)].$ --- #### Higher Dimensions >[!info] Definition >Assume an $n$-dimensional [[Itô Diffusion|Itô process]] driven by an $m$-dimensional [[Wiener Process or Brownian Motion|Wiener process]] $\mathbf{W}_t$, formally $d\mathbf{X}_t=\boldsymbol{\mu}(\mathbf{X}_t,t)+\boldsymbol{\sigma}(\mathbf{X}_t,t)d\mathbf{W}_t,$with $\mathbf{X}_t, \boldsymbol{\mu} \in \mathbb{R}^n$ and $\boldsymbol{\sigma}\in \mathbb{R}^{n \times m}$. The [[Probability Density Function|pdf]] $p(\mathbf{x},t)$ of $\mathbf{X_t}$ satisfies the Fokker-Planck equation $\frac{\partial p(\mathbf{x},t)}{\partial t}=-\sum\limits_{i=1}^{n}\frac{\partial}{\partial x_i}[\mu_i(\mathbf{x},t)p(\mathbf{x},t)]+\sum\limits_i^n \sum\limits_j^n\frac{\partial^{2}}{\partial x_i \partial x_j}{D_{ij}(\mathbf{x},t)p(\mathbf{x},t)},$where $\mathbf{D}=0.5\boldsymbol{\sigma \sigma}^{\top}$ is the diffusion [[Tensor|tensor]] with components $D_{ij}(\mathbf{x}, t)=\frac{1}{2}\sum\limits_{k=1}^{m}\sigma_{ik}(\mathbf{x},t)\sigma_{jk}(\mathbf{x},t).$