Infinitesimal generator (stochastic processes)

In mathematics — specifically, in 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^[1] that encodes a great deal of 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 L² 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.

The Kolmogorov forward equation in the notation is just ${\displaystyle \partial _{t}\rho ={\mathcal {A}}^{*}\rho }$, where ${\displaystyle \rho }$ is the probability density function, and ${\displaystyle {\mathcal {A}}^{*}}$ is the adjoint of the infinitesimal generator of the underlying stochastic process. The Klein–Kramers equation is a special case of that.