Infinitesimal Generator

Author

John Robin Inston

Published

September 25, 2026

This is the general form of transition rate matrices of finite-state continuous time Markov chains.

1 General Description

In mathematics - specifically stochastic analysis - the infinitesimal generator of a Feller process (i.e. a continuous-time Markov process satisfying certain regularity conditions) is a Fourier multiplier operator that encodes significant information about the process.

The generator is used in evolution equations such as the Kolmogorov backward equation which describes the evolution of statistics of the process; its \(L^2\) Hermitian adjoint is used in evolution equations such as the Fokker-Planck Equation, also known as Kolmogorov forward equation, which describes the evolution of the probability density functions of the process.

2 Mathematical Description

For a Feller process \((X_{t})_{t\geq 0}\) with Feller semigroup \(T=(T_{t})_{{t\geq 0}}\) and state space \(E\) we define the generator \((A,D(A))\) by \[ \begin{cases} D(A)=\left\{ f\in C_{0}(E):\lim_{ t \downarrow 0 } \frac{T_{t}f-f}{t} \text{ exists as uniform limit} \right\} \\ Af=\lim_{ t \downarrow 0 }\frac{{T_{t}f-f}}{t}, \text{ for any }f\in D(A). \end{cases} \] Here \(C_{0}(E)\) denotes the [[banarch-space|Banarch space]] of continuous functions on \(E\) vanishing at infinity, equipped with the supremum norm and \[ T_{t}f(x)=\mathbb{E}^xf(X_{t})=\mathbb{E}[f(X_{t})|X_{0}=x]. \] ## Generators of Common Processes The general \(n\)-dimensional diffusion process \[ dX_{t}=\mu(t, X_{t})dt+\sigma(t,X_{t})dW_{t}, \] has generator \[ Af=(\nabla f)^T\mu+tr((\nabla^2f)D), \] where \(D=\frac{1}{2}\sigma \sigma^T\) is the diffusion matrix, \(\nabla^2f\) is the Hessian of the function \(f\), and \(tr\) is the trace of the matrix. Its adjoint operator is \[ A^*f=-\sum_{i}\partial_{i}(f\mu_{i})+\sum_{ij}\partial_{ij}(fD_{ij}). \] The following are commonly used special cases for the general \(n\)-dimensional diffusion process:

  1. Standard Brownian Motion on \(\mathbb{R}^n\), which satisfies the stochastic differential equation \(dX_{t}=dB_{t}\) has generator \[ \frac{1}{2}\Delta \] where \(\Delta\) denotes the Laplace Operator.
  2. The two-dimensional process \(Y\) satisfying \[ dY_{t}=\begin{pmatrix}dt\\dB_{t}\end{pmatrix} \] where \(B\) is a one-dimensional Brownian motion, can be thought of as the graph of that Brownian motion, and has generator \[ Af(t,x)= \frac{{\partial f}}{\partial t}(t,x)+ \frac{1}{2}\frac{\partial^2f}{\partial x^2}(t,x). \]
  3. The Ornstein-Uhlenbeck Process on \(\mathbb{R}\) has generator \[ Af(x)=\theta(\mu-x)f'(x)+\frac{\sigma^2}{2}f''(x). \] similarly, the graph of the Ornstein-Uhlenbeck process has generator \[ Af(t,x)=\frac{\partial f}{\partial t}(t,x)+\theta(\mu-x)\frac{\partial f}{\partial x}(t,x)+\frac{\sigma^2}{2}\frac{\partial^2f}{\partial x^2}(t,x). \]
  4. A Geometric Brownian Motion on \(\mathbb{R}\) has generator \[ Af(x)=rxf'(x)+ \frac{1}{2}\alpha^2x^2f''(x). \]

3 References

  1. Schilling, R. L. and Partzsch, L. Brownian Motion: An Introduction to Stochastic Processes. De Gruyter.
  2. Math StackExchange: importance of the infinitesimal generator of Brownian motion.
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